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Linear Baire spaces and analogs of convex Baire spaces

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Linear Baire spaces and analogs of convex Baire spaces
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Todd, Aaron Rodwell, 1942-
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vii, 87 leaves. : ; 28 cm.

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Barrels ( jstor )
Increasing sequences ( jstor )
Linear transformations ( jstor )
Mathematical theorems ( jstor )
Mathematics ( jstor )
Permanence ( jstor )
Property inheritance ( jstor )
Topological spaces ( jstor )
Topological theorems ( jstor )
Topology ( jstor )
Dissertations, Academic -- Mathematics -- UF
Linear topological spaces ( lcsh )
Locally convex spaces ( lcsh )
Mathematics thesis Ph. D
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Thesis -- University of Florida.
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Bibliography: leaves 84-86.
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Typescript.
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Vita.
Statement of Responsibility:
by Aaron R. Todd.

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Linear Baire Spaces


and

Analogs of Convex Baire Spaces






By



Aaron R. Todd


A DISSERTATION PRESENTED TO THE GRADUATE
COUNCIL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF
DOCTOR OF PHILOSOPHY

UNIVERSITY OF FLORIDA 1972














To the Heroic Vietnamese People














ACKNOWLEDGEMENTS

The author wishes to acknowledge the debts owed the members of his committee, particularly his chairman, Dr. J. K. Brooks, who, perhaps without realizing it, gave much needed encouragement to the author at several strategic moments in his research. A special debt is owed by the author to his dissertation supervisor, Dr. S. A. Saxon, who suggested the work undertaken and whose criticisms helped prevent mathematical blunders and understatements of the author from reaching print.

Although not directly involved with this dissertation except for several concentrated hours of patient attention to an exposition of some of the author's discoveries, Dr. P. Bacon is gratefully recognized for a major exercise of the author's critical appreciation of mathematics through a year-long course in algebraic topology presented in the Texas style.

Lastly, the author notes the financial support for his graduate studies obtained through several pleasurable teaching appointments in the Department of Mathematics of the University of Florida, a University of Florida Graduate School Fellowship 1969-70, a National Science Foundation Summer Traineeship 1971 and the economies of his family.


iii














TABLE OF CONTENTS


Page
Acknowledgements iii

Abstract vi

Introduction 1

Section 1. Overview 2

Section 2. Some conventions, definitions

and observations 4

Chapter 1. Properties of Baire Spaces 8

Section 1. Rare and meager sets 8

Section 2. Baire spaces 14

Section 3. Almost open and open mappings 16

Section 4. Some permanence properties of

Baire spaces 18

Section 5. A Baire category theorem 21

Section 6. Pseudo-completeness of Oxtoby 23

Section 7. Productivity of the pseudocomplete property 26

Section 8. Pseudo-completeness in linear

topological spaces 28

Section 9. A productive class of convex

Baire spaces 32

Section 10. Inheritance of the Baire property 33


iv







Chapter 2. Convex Baire Space Analogs


Section 1. The new convex spaces 41

Section 2. Distinguishing examples 43

Section 3. A characterization of unordered

Baire-like spaces 48

Section 4. Some permanence properties 49

Section 5. Inheritance 53

Section 6. Productivity 56

Chapter 3. Category Analogs 64

Section 1. Analogs of meagerness 64

Section 2. The subgroup theorem and

applications 68

Section 3. Analogs of the condition of Baire 73

Section 4. An open question 75

Chapter 4. Applications 76

Section 1. Initial open mapping and closed

graph theorems 76

Section 2. Extensions of a theorem of Banach 77

Section 3. The Robertson and Robertson

theorems 80


V


41














Abstract of Dissertation Presented to the Graduate
Council of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy LINEAR BAIRE SPACES AND ANALOGS OF CONVEX BAIRE SPACES By

Aaron R. Todd

August, 1972

Chairman: J. K. Brooks
Major Department: Department of Mathematics

A locally convex linear topological space, which is barrelled, is a Baire space if and only if it is not the union of an increasing sequence of rare sets. By replacing "sets" with "linear subspaces" or "balanced convex sets" and by leaving in or removing "increasing," we obtain four distinct classes of convex spaces which fall between the convex Baire spaces and the barrelled spaces, and enjoy the following two permanence properties in addition to those known for linear Baire spaces: Each class (1) is closed under arbitrary products, and (2) contains the linear subspaces of countable codimension of each of its elements.

A contribution of this dissertation is a technique which gives the productivity property, (1), for all four classes of spaces. Also recorded are proofs of the inheritance property,

(2), for these classes. Partial results concerning these two permanence properties for the linear Baire spaces are given.


vi








In particular, productivity is established for certain classes of linear Baire spaces, and connections are discussed of the inheritance property with a long-standing and unresolved question posed by V. Klee and A. Wilansky about the null space of a linear functional in Banach space theory.

Also considered are analogs in the new spaces of sets of first and second categories which allow formulation of several theorems analogous to standard category theorems. These theorems give category-like structure in spaces which need not be Baire spaces. In particular, we obtain an analog in each of the new spaces of the subgroup theorem of S. Banach.

Applications of these concepts include refinements and generalizations of a category theorem of S. Banach on the continuous linear image of a Fr6chet space as well as a simple reformulation of the closed graph and open mapping theorems of A. P. Robertson and W. J. Robertson. In particular, this reformulation allows the replacement of convex Baire spaces in the original theorems by products of countable-codimensional linear subspaces of convex Baire spaces.


vii














INTRODUCTION

Basic to the theory of Banach spaces are the closed graph theorem and the principle of uniform boundedness. Natural to and characterized by the latter in its general formulation for locally convex topological vector spaces are the barrelled spaces. Central to the more general formulations of the closed graph theorem found in Day [1] and Robertson and Robertson [1) are the linear Baire spaces. Additionally, each locally convex Baire space is a barrelled space, and so it is natural to investigate the permanence properties of these two types of spaces as well as those of intermediate

types of spaces.

The original aim of the research recorded here was to resolve the following two questions: (1) Is the product of each family of linear Baire spaces a linear Baire space? and

(2) Is each linear subspace of countable codimension in a linear Daire space also a linear Baire space?

For "barrelled" in place of "linear Baire", the affirmative answer to the first question is well known, and the affirmative answer to the second question was established independently and through different methods by Saxon and Levin [1] and by Valdivia [1]. These observations suggest the additional aims, adopted here, of investigating the questions for several new types of convex spaces which fall


1





2

between the locally convex Baire spaces and the barrelled spaces. In following this course, the author is deeply indebted to his supervisor, Dr. S. A. Saxon, who suggested the basic definitions, and some of whose joint and independent work is necessarily discussed here.



Section 1. Overview

We generally follow the terminology and notation of

Horvath [1]; however we adopt some conventions of Robertson and Robertson [2] and of Kelley and Namioka [1]. In particular, we use "convex space" in place of "locally convex topological vector space" and "linear space" in place of "vector space".

In the chapter on analogs of convex Baire spaces, we shall find that the four new convex spaces enjoy several permanence properties in addition to those known for convex Baire spaces. In particular, a contribution of this dissertation is a technique which shows all the new properties are productive.

In the chapter on applications, it is shown, with a minor change in the proofs found in Horvath [1], that one of the four new properties, unordered Baire-like, may be used in place of the convex Baire property in the closed graph and open mapping theorems of Robertson and Robertson [1]. Although the four new properties are pairwise distinct, it is yet to be shown that the convex Baire property and unordered Baire-like property are distinct. However,






3

the affirmative answers obtained in the second chapter to questions (1) and (2) above for the unordered Baire-like property together with the new formulation of the Robertson and Robertson theorems implies that convex Baire spaces may be replaced in the original theorems by products of countable-codimensional linear subspaces of convex Baire spaces. We also find in this chapter that the unordered Baire-like property includes that infra-Baire property defined and used by Valdivia [2] in a generalization of the Robertson and Robertson closed graph theorem.

The chapter on Baire spaces is primarily introductory.

However, in the latter half, we discuss, for linear topological spaces, the concept of pseudo-completeness introduced in general topology by Oxtoby [1]. This has bearing on the question of productivity of the Baire property for a class of linear Baire spaces. Another result of Oxtoby [1] easily implies that products of separable, pseudo-metrizable convex Baire spaces are convex Baire spaces. Also considered in Lhis chapter is a partial result for the question of inheritance of the Baire property by linear subspaces of countable codimension in a linear Baire space.

In the chapter on category analogs, we demonstrate some success for concepts in the new spaces analogous to concepts of first and second category. The success is indicated by the extension to the new spaces of the Banach subgroup theorem as well as several other standard category theorems presented in Kelley and Namioka [1].





4

Section 2. Some conventions, definitions and observations

The reader will notice some ellipses in notation and

uses of alternate notations, which, once forewarned, the reader may find contribute to clarity and simplicity of expression. For example, VyFn and nlE i are used in place of l Fn and f. E. respectively, where W is the set of positive integers iEl 1
and I is an arbitrary index set.

Let E be a linear topological space. Thus E is a linear space whose scalar field is either the real or the complex numbers and which has a topology for which vector addition and scalar multiplication are continuous. E* represents the vector space of linear functionals on E, while E' is the linear subspace of E* consisting of the continuous linear functionals on E. If B is a subset of E, we say the subset of E', (f e E':lf(x)l s 1 for all x in B), is the polar of B and represent it by B0. Similarly, if C is a subset of E', the polar C0 of C is the subset of E, (x c E:lf(x)I ! 1 for all f in C1. For an element f of E*, we let N(f) represent the null space f [(0) of f. N(f) is closed in E if and only if f is continuous.

Suppose B is a subset of E. We say B absorbs a subset A of E if, for some positive real number 0, B contains aA for all Jai p. We say B is absorbing if B absorbs each singleton of E. We say B is absorbing at an element x of E if B - x is absorbing. We say B is balanced if it contains aB for all jai 1. We say B is convex if it contains (ax + (1 - a)y:O a 1, x and y in B}. We say A is






5

absolutely con7ex if it contains (ax + fy:Ial + 1 1,

x and y in B). The closure of a set with any of the above properties retains that property. A set is balanced and convex if and only if it is absolutely convex. The balanced absorbing core of a set A is the largest balanced absorbing subset of A. An absorbing set has a non-empty balanced absorbing core. The set of all finite linear combinations of elements of a set A is called the span of A and is represented by sp(A); it is the smallest linear subspace containing the set A. A balanced convex set B is absorbing if and only if, for each x in E, there is some positive integer n for which nB contains x. Thus for a balanced convex set B, we have sp(B) = UnnB.

We say B is a barrel if it is absorbing, balanced, convex and closed in E. If U is a neighborhood of 0 in E, we shall say that U is a neighborhood in E or, simply, U is a neighborhood. A subset A of the dual E' of a convex space E is equicontinuous if and only if A is a neighborhood.

A linear topological space has a local neighborhood base at 0 of absorbing balanced closed sets; a convex space has one of barrels.

A convex space has its strongest locally convex topology if and only if each absorbing balanced convex set is a neighborhood. A convex space in which each barrel is a neighborhood is said to be barrelled. A convex space E is barrelled if and only if each pointwise bounded subset of its dual E'






6

is equicontinuous. Each convex space with its strongest locally convex topology is barrelled.

R is the convex metric space of all real sequences

under term-wise convergence, or, equivalently, the product of a countably infinite family of copies of the real numbers R.

For 1 p < , is the Banach space of complex sequences

x = (ak) with the norm lxiip = (Eklaklp)1 *.

cp is the convex space of infinite countable Hamel dimension with its strongest locally convex topology. p is topologically isomorphic to the direct sum 7k Rk given the strongest locally convex topology for which each injection ik:Rk = RC*4'kRk is continuous. p is barrelled.

Suppose B is a balanced convex closed subset of a linear topological space E. B has a non-empty interior if and only if B is a neighborhood. If, in addition, E is a barrelled space, then B has a non-empty interior if and only if B is absorbing.

If a is a topology for a space X, we may write (X,U); if d is a pseudo-metric on X, we may write (X,d). We represent the closure of a set A by A~, its interior by A

Suppose (x.) is a net on a linear topological space E.

(x.). is said to be Cauchy if the net (x. - x) (ij)EIXI converges to 0 in the linear topology on E. Suppose d is a pseudo-metric on the set E, and (y ) is a sequence in E.

(y ) is said to be Cauchy in (E,d) or d-Cauchy if the net (d(ym y ))(mn9uXW of real numbers converges to zero. E is






7

said to be complete if each Cauchy net converges in the linear topology of E to an element of E. (E,d) is said to be complete or, equivalently, E is said to be d-complete if each d-Cauchy sequence converges in the pseudo-metric topology of E to an element of E. If d induces the linear topology of E, and d is translation invariant, then E is complete if and only if E is d-complete.

A linear topological space is a pseudo-metrizable

space if and only if it has a countable local neighborhood base at 0. A linear topological space which is pseudometrizable has a translation invariant pseudo-metric. A convex metrizable space which is complete is said to be a Fr6chet space. In particular, R) is a Fr6chet space.














CHAPTER 1

Properties of Baire Spaces

In this chapter we define and investigate properties of rare sets, meager sets, and Baire spaces. The most important and best known of these properties is the Baire category theorem for complete metric spaces. A scheme of Oxtoby [1] is given which covers this as well as other Baire category theorems. This scheme has a bearing on productivity of a class of Baire spaces. Simplification of these properties ard theorems in the setting of linear topological spaces are given, and the chapter ends with a discussion of an open question about inheritance of the Baire property in linear topological spaces.



Section 1. Rare and meager sets

We begin with a definition and an elementary, but frequently used, observation.



1.1 Definition. A subset A of a topological space X is dense in a subset B of X if and only if its closure A contains B. A set dense in X is said to be dense.



1.2 Proposition. If a subset A of a topological space is dense in an open set U, then their intersection U n A is


8






9

dense in U; moreover (U n A) = U



Proof. Clearly (U n A) is contained in U , and so we need only show that (U n A) contains U. Suppose x is in U, and N is any neighborhood of x. Since x c U C A , and N n U is a neighborhood of x, N n U meets A. Thus N meets U n A, and so x is in (U n A).//



1.3 Definition. A subset A of a topological space X is rare in X, or, simply, rare, if and only if its closure A contains no non-empty open set of X.



A term often used for rare is nowhere dense. An example of a rare subset of a topological space is a proper closed linear subspace F of a linear topological space E. For if F contains a non-empty open set U, then, with x in U, U - x is contained in F and is a neighborhood in E. Thus E = U nn(U - x) C F C E, and so E = F.

If a set A is not rare, then its closure contains a nonempty open set U, and so A is dense in the non-empty open set U. The converse holds as well. The following records this characterization of non-rare sets.



1.4 Proposition. A subset of a topological space is not rare if and only if it is dense in some non-empty open set.






10

1.5 Definition. A subset A of a topological space X is rare in a subset B of X if and only if their intersection A n B is rare in the subspace B.



1.6 Definition. A subset A of a topological space X is meager in a subset B of X if and only if A is contained in a countable union U A of subsets A of X, each rare in B.
n n n
A set meager in X is said to be meager.



We observe that a set A is meager in a set B if and only if A n B is meager in the subspace B.

A meager set is also said to be of the first category, while a non-meager set is of the second category. Several elementary properties of meager sets follow from corresponding properties of rare sets. For this reason they are given together in the following three propositions.



1.7 Proposition. If a rare (meager] subset of a topological space contains a set A, then A is rare [meager].



Proof. Suppose a set B contains A. If BF has empty

interior, then the subset A of B_ has empty interior, and so A is rare. If B is contained in U A with each A rare, n n n
then clearly, A is contained in U A as well, and so A is nmn
meager .//






11

1.8 Proposition. If a set A is rare [meager] in a set B, then their intersection A n B is rare [meager].



Proof. We may assume A is contained in B. Suppose A is dense in an open set U. If A is rare in B, then, since U n B is contained in A- n B, we have U n B empty, and so U n A is empty. From proposition 2, U (U 0 A) =, and

so U is empty. Hence A is rare.

If A is meager in B, then A is contained in some countable union U A with each A rare in B. Thus each A n B
n n n n
is rare. Now A n B is contained in U (A n B) and so is n n
meager.//

A rare set need not be rare in itself. For example


a singleton

in itself. of the rare


of the real line is rare, but is not even meager

The next three propositions concern inheritance and meager properties.


1.9 Proposition. A subset A of a topological space is rare [meager] in an open set U if and only if A n U is rare [meager].



Proof. Proposition 8 gives one direction for both properties. Conversely, let A n U be rare, and suppose V is an open set. Now if the open set V n U of subspace U is contained in A n U, then A is dense in V n U. Since V n U is open in the space, proposition 2 gives V nUC (V nU nA) C (U n A). Yet U n A is rare, so V n U is empty. Thus A is rare in U.






12

If A q U is contained in U A with each A rare, then by nfn n
proposition 7, each An n U is rare. By the above, each A is rare in U. Also, A\U is rare in U. Since A is contained in (A\U) U (Un A n), A is meager in U.//

From this, we easily see that a meager s.et is meager in every open set. With additional restrictions a meager set is meager in itself. As the following shows, these two properties are equivalent for dense sets.



1.10 Proposition. If a set A is dense in a topological space, then A is meager in itself if and only if A is meager.



Proof. If A is meager in A, then, by proposition 8, A is meager. Conversely, suppose A is contained in U A with n n
each A rare, and let U be an open set. If U n A is contained in A fl A, then, since A is dense in U, we have U C (U q A) C (An n A) C A . Since An is rare, U is empty. Thus U n A is empty, and so An is rare in A. Hence A is meager in itself.//

We may localize this property as shown in the following.



1.11 Corollary. If a set A is dense in an open set U, then A is meager in A n U if and only if A is meager in U.



Proof. By proposition 2, A q U is a dense subset of subspace U. By the above proposition, A n U is meager in itself if and only if A n U is meager is subspace U.






13

Thus A is meager in A n U if and only if A is meager in

U.//



1.12 Proposition. The collection G of rare [meager] subsets of a topological space is closed under intersections and finite [countable] unions.



Proof. Since a subset of a rare [meager] set is rare [meager], G is closed under intersections. If A and A' are rare, and A U A' is dense in an open set U, then U is contained in (A U A') A j A' . Now the open set U\A' is contained in the rare set A , and so U\A' is empty. Thus U is contained in the rare set A' . Hence U is empty, and A U A' is rare. By induction, a finite union of rare sets is rare. Suppose A C U A with each A rare. Now U A n m mn mn n n
is contained in the countable union U mnA , and so the countable union Un An of meager sets is meager.//

This proposition provides a convenient characterization of meager sets.



1.13 Proposition. A subset of a topological space is meager if and only if it is the union of an increasing sequence of rare sets.


Proof. If A is contained in U A with each A rare, then n n n
A equals Un(A1U ... U An) flA. By proposition 12, A U ... UAn is rare, and so (A1U ... UAn) fA is rare. The converse is clear.//






14


Section 2. Baire spaces

The next definition follows Bourbaki [2].



1.14 Definition. A topological space X is a Baire space if and only if each non-empty open subset of X is non-meager.



There are several convenient characterizations of the

Baire property which use the observation that the complement of an open dense set is a closed rare set and vice-versa.



1.15 Proposition. For a topological space X, each of the following is equivalent to the Baire property:

(i) Each countable intersection of dense open sets is dense.

(ii) The complement of each meager set is dense.



Proof. If (U ) is a sequence of dense open subsets of the Baire space X, then each A = X\Un is closed and rare. If U is a non-empty open set, then U is not contained in the meager set U A . Thus U n (n u) = U n (n (X\A )) =
n n n n n n
U\U nA n , 0, and so n U is dense.

For (i) implies (ii), suppose A is contained in Un An

with each A rare in the topological space X. Each U = X\A is a dense open set. Now X\A D X\U A D X\U A = n (X\A ) nn n n n n
nn U . By (i), nn U is dense, thus X\A is dense. n n n n






15

Finally, suppose (ii) holds, and U is a meager open set of a topological space X. Now X1\U is dense, yet it is disjoint from the open set U. Hence U is empty, and X is a Baire space.//

The Baire property satisfies several permanence properties which we investigate in this and following sections. Proofs of the following proposition may be based on proposition 15. The one appearing here was chosen since it uses many of the foregoing propositions.



1.16 Proposition. If a topological space X contains a dense subspace Y which is a Baire space, then X is a Baire space.



Proof. Suppose U is a meager open set of X. Now Y n U is meager, and so, since U is open, Y Q U is meager in the subspace U by proposition 9. Since Y is dense in the open set U, Y n U is dense in U by proposition 2. Hence Y n U is meager in itself by proposition 10. Now Y n U is meager in Y by proposition 9. But Y is a Baire space, and Y q U is open in Y, thus Y n U is empty. Since Y is dense and U is open in X, U is empty. Thus X is a Baire space.//

Certain subspaces of a Baire space are themselves Baire spaces. The following gives an example.


1.17 Proposition. The Baire property is open hereditary.






16

Proof. Let U be an open subset of a Baire space X. If V is open in X, and V n U is meager in U, then the open set V n U is meager in X. Since X is a Baire space, V n U is empty. Thus U is a Baire space.//



Section 3. Almost open and open mappings

As shown in Frolik [2], several types of mappings preserve the Baire property. However, we shall only be concerned with ones which arise naturally in linear topological spaces.



1.18 Definition. A mapping f from a topological space X into a topological space Y is almost open if and only if, for each point x of X and each neighborhood N of x, f[N] n f[XJ is a neighborhood of f(x) in the subspace f[X] of Y.



1.19 Proposition. If f is a mapping from a topological space X onto a topological space Y, then f is almost open if and only if f[U] is contained in f(U]' for each open set U of X.



Proof. Let U be an open set of X. For each x in U, f[U] is a neighborhood of f(x), and so f(x) is in f[U]~i. Thus f[U] is contained if f[U] . Conversely, let x be in X, and N be a neighborhood of x. Now x is in the open set N ,
i i i -i - 1
so f(x) is in f[N. Yet f[N ] C f[N] C f[N] . Thus

f[N] is a neighborhood of f(x).//






17

1.20 Definition. A mapping from a topological space X into a topological space Y is an open mapping if and only if for each open set U of X, f[U] is an open set of the subspace f[X] of Y.



From the characterization of an almost open mapping, we see that an open mapping is almost open. An example of an open mapping is the canonical mapping of a linear topological space onto its quotient by a linear subspace. .

The restriction of a continuous open mapping may not be an open mapping even if the mapping is, in addition, a linear mapping which is restricted to a dense linear subspace of a normed space. This is the crux of the following.



1.21 Example. Let M be a proper dense linear subspace of an infinite dimensional Banach space B with N a closed linear subspace which is an algebraic supplement to M, that is, B = M E N. Such subspaces exist. For example, let M be the null space of a linear function g, discontinuous on B, and N be sp([x)) where g(x) Z 0.

Let f be the canonical mapping of B onto the quotient space B/N. Since N is a closed linear subspace of a Banach space, B/N is a Banach space and so is complete. Now f is a continuous open linear mapping, yet the restriction of f to M is not open. Otherwise, since g = fjM is injective and continuous, g would be a topological isomorphism of the incomplete space M with the complete space B/N.//






18

Almost open mappings are better behaved in this respect.



1.22 Proposition. If f is a continuous almost open mapping from a topological space X into a topological space Z, and Y is a dense subspace of X, then f restricted to Y is an almost open mapping from Y into Z.



Proof. Let x be in Y, and U be an open neighborhood of x in X. We need to show f[U n Y] n f[Y] is a neighborhood of f(x) in f[Y]. By continuity of f and proposition 2, f[U n Y] D f[(U q Y)} D f[U], so that f[U n Y] contains f[U] . Now, since f is almost open, f[U] q f[X] is a neighborhood of f(x) in f[X], and so flU] n f[Y] is a neighborhood of x in f[Y]. Yet f[U n Y] n f[Y] contains f[U] f f[Y], and so is a neighborhood of x in f[YI.//



Section 4. Some permanence properties of Baire spaces

In this and following sections, we consider permanence properties of the Baire property.



1.23 Proposition. The Baire property is preserved by continuous almost open mappings.



Proof. Let f be a continuous almost open mapping from a Baire space X onto a topological space Y. If A is rare in Y, then f 1[A] is rare in X. For suppose f~ [A] is dense






19

in an open set U of X. By continuity,

f(U] C f[f I[A]-] C f[f1 [A]]- = A-,

and so f[U] is contained in A7. Since f is almost open, proposition 19 gives f[U] C f[UF . Thus, if U is nonempty, f(U] is not rare. Yet f[U] is a subset of the rare set A. Hence U is empty.

Now if V is an open set of Y contained in U A with n n
each A rare in Y, then f 1[V] is contained in Un [A] with each f~ [An ] rare in X. Thus f 1[V] is meager in the Baire space X. Since f is continuous, f 1[V] is open in X, so f1 [V], and therefore V, is empty. Hence Y is a Baire space.//



1.24 Corollary. If f is a continuous almost open mapping from a topological space X into a topological space Z, and Y is a dense Baire subspace of X, then f[Y] is a Baire space.



Proof. From proposition 22, the restriction g of f to Y is almost open. Since g is continuous, the above proposition shows that f[Y] = g[Y] is Baire space.//



1.25 Corollary. If a linear topological space E is a Baire space and F is a linear subspace of E, then the quotient space E/F is a Baire space.


Proof. The canonical mapping f from E onto E/F is a continuous open mapping, and the proposition shows that E/F = f[E] is a Baire space.//






20

Under certain conditions on a non-empty topological

space, for example homogeniety, the Baire property is equivalent to non-meagerness of the space. In particular and for simpler reasons, this equivalence holds for linear topological spaces.



1.26 Proposition. A linear topological space is a Baire space if and only if it is not meager in itself.



Proof. Let E be a linear topological space. If E is a Baire space, then the non-empty open set E is not meager. Conversely, suppose U is a non-empty open set. If x is in U, then U - x is a neighborhood in E, and so is absorbing. Thus E = U nn(U - x). If E is non-meager, then by proposition 12, some n(U - x) is non-meager. But f(y) = (1/n)y + x for each y in E is a homeomorphism of E, so U f[n(U - x)] is non-meager, and E is a Baire space.//

The next corollary characterizes linear topological spaces which are Baire spaces. It serves as a model for defining the new types of spaces studied in the following chapters. The proof follows immediately from the characterization of meager sets in proposition 13.



1.27 Corollary. A linear topological space is a Baire space if and only if it is not the union of an increasing sequence of rare sets.






21

1.28 Corollary. A dense linear subspace F of a linear topological space E is a Baire space if and only if F is not meager in E.



Proof. From the proposition, F is a Baire space if and only if F is not meager in F. Since F is dense, proposition 10 shows that F is non-meager in itself if and only if F is not meager in E.//



Section 5. A Baire category theorem

Concerning existence of non-meager sets, Kelley and

Namioka [1, p. 85] state: "In spite of a great deal of work on the subject, there is essentially only one method known for showing that a set is of the second category." Cullen [11, for instance, records four Baire category theorems, each essentially proved by the construction of a convergent sequence. However, only relatively recently have inclusive techniques been published. Such techniques are found in De Groot [1], FrolCk [1], and Oxtoby [1]. The last technique is reflected in the proof of the following classic Baire category theorem.



1.29 Theorem. A complete pseudo-metric space is a Baire space.



Proof. Suppose (U ) is a sequence of open dense subsets of the space. Let U be a non-empty open set. We shall show that nun meets U, and so n n U is dense. Thus, from proposition 15, the space is a Baire space.






22

Denote by en the collection of all non-empty open

balls Vr (x) = (y e X:d(x,y) < r), x E X, 0 < r < 1/n, of the complete pseudo-metric space (X,d).

If V is a non-empty open set, then there is an element B of Rn with closure B contained in V. For suppose x is in V, then for some 0 < r < 1/n, V (x) is contained in V. Let B = V (x). Now B is in ni and is contained in the
r/ 2 n
closed set

(y E X:d(x,y) ! r/2} C Vr (X) C V.

Thus B is contained in V.

If B is in A with B D B , then ni B is non-empty.
n n n n+l' n n
For let x be in B . With m s n, d(x ,x ) < 2/m, and so
n n m' n
(x ) is a Cauchy sequence. By completeness, (x n) converges to some element x of X. Since Bm D Bn for m n, dist(B ,x) >r, for some positive r and some m, implies d(x ,x) > r for all n m, a contradiction. Hence dist(B mx) = 0 for all m, and so x is in each B . Therefore x is in ni B = nn B , and
m n n n n fn B is non-empty.

For notational convenience, let B0 U. Since U is a
0 1
dense open set, B0 n U1 is a non-empty open set. Choose B1 in 61 with B contained in B0 n U1. Suppose Bk k"

1 s k s n, are chosen with B C Bn-1 n U . Since Un+1 is a dense open set, B ni Un+1 is a non-empty open set, and there is B E 0 with B7 C B n U Thus we have a sequence
n+1 n+1 n+1 n n+l
of terms B n c with B D B ni U D B , so that nn U
n n n n n+ n+ n n
contains the non-empty set finB C B0 = U. Thus fi Un meets U.//






23

Section 6. Pseudo-completeness of Oxtoby

De Groot [11 has shown that the property of subcompactness defined by him together with regularity implies the Baire property. His proposition has, as corollaries, the standard Baire category theorems. Furthermore, he has shown that subcompactness for a metrizable space is equivalent to existence of a metric which induces the topology and for which the space is complete. We show that the pseudo-completeness property of Oxtoby [1] does not have this equivalence. We note here that the property of pseudocompleteness is satisfied by the hypotheses of the standard Baire category theorems and is implied by subcompactness with regularity.



1.30 Definition. A topological space X is quasi-regular if and only if each non-empty open set contains the closure of

a non-empty open set.



1.31 Definition. A collection R of non-empty open sets of a topological space X is a pseudo-base for X if and only if each non-empty open subset contains an element of Q.



1.32 Definition. A topological space X is pseudo-complete if and only if it is quasi-regular and there is a sequence

(13) of pseudo-bases for X such that nl B is non-empty if
n n n
B E 6 and B D B , n E W. n n n n+l






24

It is clear that, in contrast to completeness for a pseudo-metric space, pseudo-completeness is a topological invariant.



1.33 Proposition. A complete pseudo-metric space is pseudocomplete.



Proof. As in the proof of theorem 29, denote by a the

collection of non-empty open balls Vr (x) = (y E X:d(x,y) < r), for x e X, 0 < r < 1/n, of a complete pseudo-metric space (X,d). For a non-empty open set V, we found an element B of an with B contained in V. Thus X is quasi-regular, and each n is a pseudo-base. Moreover, for B n E3 with
n n n
B D B , we showed that fl B is non-empty. Thus X is n n+l' n n
pseudo-complete.//

A metrizable space X is said to be topologically complete if there is a metric d inducing the topology of X for which (X,d) is a complete metric space. A set is said to be a G -set if it is the intersection of a countable family of open sets. A metrizable space X is said to be an absolute G -set if X is a G -set in every metrizable space in which
-6
it is embedded as a subspace. It is an exercise in Kelley [11, that a metrizable space is topologically complete if and only if it is an absolute G -set.

The following example shows that a pseudo-complete metric space need not be topologically complete.






25

1.34 Example. Let X C R be the union of the open upper half plane S = R X (0,r-) and the rational points Q X (0) on the boundary of S. Let X have the subspace topology induced by the Euclidean metric d on R . We shall show that X is pseudo-complete, but not a G -set of R , and so, not topologically complete.

Let Vr (x) = (y E X:d(x,y) < r), for x c X, r > 0, and let 3n = (V (x):x G Q X (O,w), 0 < r < 1/n, S contains V (x)}. We note that if V (x) is in I , then its closure r rn
Vr (x) is contained in X. Thus for Bn n with B DBn+1 fx. we have B D B n+l and so n B is non-empty as shown in the
n n1n n
proof of theorem 29.

We need to show X is quasi-regular and each an is a

pseudo-base for X. If an open set U of R meets X, then U
2
meets the open set S, since X C S . But Q X R is dense in R , so Q X R meets U n S. Suppose x is in (Q X R) n U n SC Q x (oa), then there is 0 < r < 1/n with

U n X D U n S D vr(x) D (y c R2:d(x,y) f r/2) D Vr/2

and so U n X contains the closed set Vr/2(X) D r/2(x). Thus X is quasi-regular, and, since V r/2(x) is in n , fn is a pseudobase for X. Hence X is pseudo-complete.
2
If X is topologically complete, then X is a G -set of R Hence Q x (0) = X n (R x (0)) is a G -set of the subspace
6
R X (o), and so Q is a G -set of R. If Q = nn U where each Un is open in R, then U is dense in R, and R\Un is rare, and so meager. Q is a countable union of singleton sets, and so is meager. Thus U n(R\U ) U Q is meager. Yet






26

Un (R\Un) U Q = (R\n U) U Q = R, and so R is meager, a contradiction.//



1.35 Theorem. A pseudo-complete space is a Baire space.



Proof. Let V be a non-empty open subset of a pseudocomplete space X with (B ) as in definition 32. From quasiregularity, there is a non-empty open set W with closure W contained in V. Since 8n is a pseudo-base, there is a B in a with B contained in W. Thus B C W7 C V. The remainder of the proof consists of the first and last paragraphs of the

proof of theorem 29.//



Section 7. Productivity of the pseudo-complete property

Assuming the continuum hypothesis, Oxtoby [1] gives an example of a completely regular Baire space whose product with itself is meager. An open question cited there is the following: Is the product of two metrizable Baire spaces a Baire space? We note the question: Is the product of two or more linear Baire spaces a Baire space?

In contrast to the Baire property, pseudo-completeness is productive. With slight changes we record the proof of Oxtoby [1].



1.36 Theorem. A product of pseudo-complete spaces is pseudocomplete.






27

Proof. Suppose {X.3 are pseudo-complete spaces and for each i in I, (a )n is a sequence of pseudo-bases for X satisfying definition 32. We may suppose X. is in each B ..
1 fni
Define n = (I B.:B. c B ., B. = X. for all but a finite
n I i 1 nll 1 1
number of i in I), so that Bn is a collection of non-empty open sets of the product space X = lIX .

If V is a non-empty open subset of X, then there is an open set EIU . contained in V with each U a non-empty open subset of X., all but a finite number of which equal X.. If
1
U. Xi, let B. = U., otherwise, choose B. in B . with closure
1 1' 1' ni
B. contained in U.. Thus H B. is in B , and T B.C (TI B.) = 1I i n I i II
B C '7 U C V. Hence X is quasi-regular, and each n is a I . I i n
pseudo-base for X.

Suppose B E with EI B . = Bn D B n+ I B )

SB ..* For each i, B. EB . and B . D B~ ., so that I n+l,i ni ni ni n+l, i
n B . P. Now = B (IB .) = T ( B .) / ). Thus X n ni n n n ni I n ni
is pseudo-complete.//

From this and theorem 35, it is clear that a product of pseudo-complete spaces is a Baire space. In particular, we have the following well-known result.



1.37 Corollary. A product of complete pseudo-metric spaces is a Baire space.



Proof. By proposition 33, a complete pseudo-metric space is pseudo-complete. Thus, from the theorem, a product of complete pseudo-metric spaces is pseudo-complete, and so a Baire space by theorem 35.//






28

Section 8. Pseudo-completeness in linear topological spaces

In the next theorem, which gives a restriction on the

utility of pseudo-completeness for linear topological spaces, we shall use a corollary of a well-known difference theorem discussed in chapter 3. It states that the difference set A - A of a non-meager Borel subset A of a linear topological space is a neighborhood.



1.38 Theorem. A linear metrizable space is pseudo-complete if and only if it is complete.



Proof. Let (E,d') be a linear metric space with translation invariant metric d'. If E is complete, then E is complete in the metric d'. Proposition 33 shows that E is pseudo-complete. Conversely, suppose (an ) is a sequence of pseudo-bases which satisfies the definition of pseudo-completeness. Let F be the completion of E, and d be the extension of d' to F. Now E is dense in F, and (F,d) is a complete metric space. We shall show that E equals F.

Define C = (C: C is open in F, C n E E s and
mn n
diam(C) < 1/m). Let Umn = Ucmn, and U be a non-empty open subset of F. For x in U, there is 0 < r < 1/(2m) with V r(x) contained in U. Since ! is a pseudo-base for E, and E is dense in F, there is an element B of 5 n contained in the non-empty open set of E, Vr (x) n E. Let C be an open set of F with B = C f E C V r(x). Since E is dense in the open set C, C C (C n E) C V r(x) C U. Also C C Vr (x) C





29

(y c F:d(x,y) s r}, and so diam(C) s 2r < 2(1/(2m)) = 1/m. Hence C is in Cmn and so C is contained in (UC = U .
mn mnmn Now P 7' B C C C U nlU , so U n is a dense open set of F.

Let A = ni U . Now F\A = F\i U = U (F\U ) with
inn mn mn inn mn in
each F\Umn rare in F, so that F\A is meager in F. But

F = AU (F\A) is a Baire space, so, by proposition 12, A is non-meager in F.

Now A is a G -set and so a Borel set. From the remark
6
preceding the theorem, the difference set A - A is a neighborhood in F. We need only show that A is contained in E, for then E C F = U nn(A - A) C E, and so E = F.

Let x be in A. For each m and n, choose Cmn in Cmn containing x. For each m and n, there is m' > m with C rnD C .n+i For assume not, then let r = dist(xF\C n)

Since Cmn is open and contains x, r is positive. For all M' > M, x is in Cmn n Cm'n+1 and, by assumption, C,'n+l\Cmn is non-empty, and thus we have 0 < r - diam(C )'n+) Yet diam(C' ) = diam(Cm'n+1),< 1/m'. Hence 0 < r < 1/m', for all m' > m, a contradiction.

We may thus select a strictly increasing sequence (mn) with C e C and C D C . Let B C n E .
m n m n m n m ln+l n m n n
n n n +1n

Now E is dense in the open set Cmkk' so Bk = (Cmkk n E) C by proposition 2. Thus B = C n E D C n E
mk n inn m +1n+

B +l n E. By the pseudo-completeness of E, we know ni B is a non-empty subset of E contained in n C n. But x is in
n





30

nl C , the sequence (m ) is increasing, and diam(C ) < n mn n mnn

1/mn. Thus n C = (x3, and so {xj = n B C E. Hence
nn mn n n
n
A is contained in E.//

Suppose a linear metrizable space E is topologically

complete. As a subspace of its completion F, E is a G -set. But E is a Baire space and dense in F, so by corollary 28, E is non-meager in F. Hence E = E - E is a neighborhood in F, and so E = F. Thus E is complete. By proposition 33, we may obtain this well-known result as a corollary of the above theorem, and the proof of the following is clear.



1.39 Corollary. A linear metrizable space E is topologically complete if and only if E is complete if and only if E is pseudo-complete.



In example 34, we described a pseudo-complete metric

space which was not topologically complete. The following, essentially an exercise of Kelley and Namioka [1], givesexistence of incomplete linear topological spaces which are, nonetheless Baire spaces. In particular, there are incomplete normed spaces which are Baire spaces. By the corollary and theorem, such normed spaces are neither pseudocomplete nor topologically complete.



1.40 Example. Let E be a linear topological space which is a Baire space and has infinite dimension. An infinite dimen-






31

sional Banach space suffices. Suppose H is a Hamel basis for E. Choose and index a countably infinite subset [xn)n of H, and let En sp(H\x n1).

Since each element of E is a finite linear combination of elements of H, and (x nIn is a countably infinite set, we have E = UnEn. From proposition 12, and since E is a Baire space, some En is not meager in E. Thus En is not rare, and as En is a linear subspace of E, it is dense in E. By corollary 28, E dense and not meager implies En is a Baire space. But En is a proper dense subspace of E, and so E is incomplete.//

It is well known that a complete convex space need not be a Baire space, example 2.8 provides just such a space. Thus, by theorem 35, there are complete convex spaces which are not pseudo-complete.

The product of an uncountable family of non-trivial

Frechet spaces is not pseudo-metrizable, yet, by proposition 33 and theorem 36, such a product is pseudo-complete. Thus, there are pseudo-complete convex spaces which are not pseudometrizable. A more interesting example, which, by theorem 38, implies this, would be an incomplete linear space which

is pseudo-complete.

It is easy to see the closure of a pseudo-complete subspace is pseudo-complete, and, with theorem 38, a one-codimensional linear subspace of a pseudo-complete linear space need not be pseudo-complete.






32


Section 9. A productive class of convex Baire spaces

There is yet another result of Oxtoby [11 which may be usefully specialized to convex spaces. We record it here.



1.41 Theorem [xtoby]. The product of any family of Baire spaces, each of which has a countable pseudo-base, is a Baire space.



A convex space with a countable pseudo-base has a particularly simple characterization.



1.42 Proposition. A convex space has a countable pseudobase if and only if it is a separable pseudo-metrizable space.



Proof. One way is clear. For the other, suppose B is a countable pseudo-base for a convex space E. By choosing one point in each element of 3, we obtain a countable dense set, and so E is separable.

To complete the proof, we obtain a countable neighborhood base at 0 from f. For a set B, define bc(B) to be the smallest balanced convex set containing B. Let M = (bc(B): B E 91. Suppose B is in 2, and x is in B. Now the open set V = (1/2) (-x) + (1/2)B contains 0 and is contained in bc(B). Hence the countable family M is a family of neighborhoods of 0.






33

For a balanced convex neighborhood U, there is an element B of R contained in U. Hence bc(B) is contained in U, and T is a neighborhood base at 0.//

Clearly, theorem 41 reduces to the following for convex spaces.



1.43 Theorem. A product of any family of convex Baire spaces, each a separable pseudo-metrizable space, is a Baire space.



Section 10. Inheritance of the Baire property

A long-standing problem noted by N. J. Kalton, who attributes it to V. K. Klee and A. Wilansky, is to prove or disprove the following:



1.44 Conjecture. A non-zero linear functional on a Banach space is continuous if and only if its null space is meager.*



Of course, it is well known that a linear functional on a linear topological space is continuous if and only if its null space is closed. This may be recast in slightly different terms. Since the null space of a non-zero linear functional is a linear subspace of codimension 1, it is closed if and only if not dense, and not dense if and only if rare. Thus a non-zero linear functional is continuous if and only if its null space is rare. This is related to the conjecture above, and both are connected to the following.


*
N. J. Kalton, personal communication to S. A. Saxon






34

1.45 Conjecture. A non-zero linear functional on a linear Baire space is continuous if and only if its null space is meager.



As will be shown, this conjecture is equivalent to inheritance of the Baire property by linear subspaces of countable codimension. We begin with the following observation.



1.46 Proposition. Conjecture 45 is equivalent to inheritance of the Baire property by dense linear subspaces of codimension 1.



Proof. Let E be a linear Baire space. Suppose the

conjecture holds, and N is a dense linear subspace of codimension 1 in E. By corollary 28, we need only show that N is non-meager in E in order to show that N is a Baire space.

Let x be in E\N, so E = N e sp({x)). Define a function

f from N U (x) to the scalar field by f[N] = (0), and f(x) = 1, and extend f to E by linearity. Now f is a linear functional on E with null space N. Since N is a dense proper subspace of E, f is not continuous. By the conjecture, the null space N is non-meager.

Conversely, the null space of a non-zero continuous

linear function on E is a proper closed linear subspace of E. Such a linear subspace is rare, and therefore meager. Assuming






35


the inheritance property, we need only show that a discontinuous linear functional f on E has a non-meager null space. Let N = N(f) be the null space of f. Since f is discontinuous, N is dense. But N is also of codimension 1, and so is a Baire space. Corollary 28 shows that N is nonmeager.//

The following gives a technical observation.



1.47 Lemma. Let the linear topological space E contain linear subspaces F and G with intersection {0]. Suppose F and G contain a closed set A and a compact set K, respectively.

If A is rare in F, then A + K is rare in E.



Proof. Suppose A + G is a neighborhood of some point in E. That is, for some a in A and g in G, A + G is a neighborhood of a + g. Clearly, (A + G - g) n F is a neighborhood of a in F. For x in (A + G - g) n F there are a' in A and g' in G with x = a' + g' - g E F. Hence x - a'= g' - g is in F n G = (0), so that x = a' e A. Thus (A + G - g) n F is contained in A, and so is a neighborhood of a in F.

Now A is closed in E and K is compact, so A + K is closed in E. If A + K is not rare in E, then A + K and so A + G is a neighborhood of some point of E. From the foregoing, A is a neighborhood of some point in F. So A is not rare in F, and the contrapositive gives the lemma.//






36

The following example shows that the requirement in the lemma that A be closed in E is important.



1.48 Example. Let E be an infinite dimensional normed linear space with closed linear subspace L of codimension 1. Suppose H is a Hamel basis for L with each element of norm 1, x is in E\L, and (x ) a sequence of distinct elements of H. Define a scalar-valued function f on H U [x) by n; u = x
f(u) =
0; otherwise,

and extend f to E = sp(H U [x)) by linearity. Now f is a linear functional on E. Since (xn) is contained in the linear subspace L, we observe that xn /n converges to 0 in L, since ixn/n1 =xn /n = 1/n converges to 0, Yet f(x /n) f(x n)/n = n/n = 1 does not converge to 0. Thus f is discontinuous on L, and so on E. From this, the null spaces N(f) and N(flL) = L n N(f) are proper dense linear subspaces of E and L respectively.

Let F = N(f). It is easy to show that E\(F U L) is

non-empty, so let y be in E\(F U L). Now the intersection of linear subspaces F and G = sp([y)) is [0), in fact, E = F G G. Let A = L n F C F, and K = (ay: jai s 1) C G, so that A is closed in F and K is compact.

Since L is a closed linear subspace of E, A = L F is a closed linear subspace of F. If A = L n F = F, then






37

F C L. But F = N(f) is dense and L is closed in E, so L = E. This is a contradiction since L is of codimension 1. Thus A = L n F is a proper closed linear subspace of F, and so A is rare in F. However A + K is not rare in E as is now shown.

Since A7 is closed in E and K is compact, A + K is closed in E. From this and continuity of addition,

A + K D (A + K) D A + K = A + K,

so (A + K) = A + K = (L n F) + K = L + K, which is a weak neighborhood of 0. For define g[L] = {0), and g(y) = 1 and extend g to E = L E G by linearity. Since the null space N(g) = L of g is closed, g is continuous, moreover, (A + K) = L + K = (u e E: Ig(u)] : s 1). Hence A + K is not rare in E.//

We recall that an F -set of a topological space is the union of a countable family of closed sets.



1.49 Theorem. The Baire property is inherited by F -linear subspaces of countable codimension.



Proof. Let E be a linear Baire space. Suppose F is a proper linear subspace of countable codimension in E, and F is an F -set of E. We shall show that F is nona
meager in itself. Let (A ) be an increasing sequence of closed subsets of F with F = U An





38

Since F is an F -set of E, there is an increasing sequence (C n) of closed subsets of E with F = U Cn* Now F = U (A nn C ). Also, since C C F is closed in E
n n n 'n
and A C F is closed in F,
n
(A n c )C A n C= n c =A n (C nF) =(A nF) n c
n n n n n n n n n n

= A flC,
Sn n

so A n n is closed in E. Finally, if A nf C is not rare in F, then A is not rare in F. For notational convenience, we may assume each An is closed in E.

Since F is a proper linear subspace of E and of countable codimension, there is a sequence (xn) in E such that E = F S sp([x n) ). Letting G = sp([x ) ), we have Fn G= (0). Each A is contained in F and closed in E; each Kn
nn
(Elakxk: Ick n for 1 s k s nj is contained in G and compact. Now E = Un (A + Kn ), and E is a Baire space. Thus some A + Kn is not rare in E, and so, by the lemma, An is not rare in F.//

It is not difficult to see, independent of the above, that a closed linear subspace of countable codimension in a linear Baire space is of finite codimension. This is included in the following.



1.50 Corollary. An F -linear subspace of countable codimension in a linear Baire space is closed and of finite codimension.






39

Proof. Suppose F is a linear subspace of a linear Baire space E, and (x n) is a sequence in E with E = F + sp([xn n). If F is closed, then each E = F + sp({xjln) is closed. Yet E = InE . As E is a n 1l n n
Baire space, some En is not rare. Since a proper closed linear subspace is rare, E = En F + sp({xk) ), and so F is of finite codimension in E. We need only show that F is closed if F is an F subset of E.

Let (C ) be a sequence of closed subsets of E with F = U C . From the theorem, F is a Baire space, so that
n n
some Cn is not rare in F. Thus for some open set U, C n f F contains U n F 7# 4. But F is dense in the open
n
set U q F of the space F , so by proposition 2, (U n F) contains U F. Now F DC = CD (U n F) :DU F .
n n
Since U n F is a non-empty open subset of F, F is F , and so closed.//

Finally, we have the following.



1.51 Proposition. Conjecture 45 is equivalent to inheritance of the Baire property by linear subspaces of countable codimension.



Proof. Let F be a linear subspace of countable codimension in a linear Baire space E. Let (x ) be a sequence in E with E = F + sp((x )n). Letting E = F + sp({xk)n), we have E = U E . Since E is a Baire space, some E is
n n n
not meager in E. By corollary 28, En is a Baire space.





40

Now F is of finite codimension in En = F + sp([xk) ). Also F n En is a closed linear subspace of Baire space E and of finite codimension in E . Hence F n E is a n n n
Baire space by theorem 49. Since F is of finite codimension in F n En and dense in F n E n we may assume that F is of finite codimension in E and dense in E.

From finite induction, inheritance of the Baire property by dense finite codimensional linear subspaces is equivalent to inheritance by dense 1-codimensional linear subspaces. Proposition 46 completes the proof.//














CHAPTER 2

Convex Baire Space Analogs

In this chapter we generally restrict our attention to convex spaces, and consider for these spaces several properties related to the Baire property. Relations among these properties are discussed, and permanence of these properties under certain operations is found to be stronger than presently known for the Baire property itself. In particular, the properties discussed are productive and inherited by linear subspaces of countable codimension, both permanence properties which are, as yet, unknown for the Baire property.



Section 1. The new convex spaces

From corollary 1.27, we obtain the following observation: A convex space is a Baire space if and only if it is not the union of an increasing sequence of rare sets. Using this characterization as a model, we may obtain more general properties which we now define.



2.1 Definition. A convex space is Baire-like if and only if it is not the union of an increasing sequence of balanced convex rare sets.


41






42

2.2 Definition. A barrelled space is quasi-Baire if and only if it is not the union of an increasing sequence of rare linear subspaces.



By relaxing the inclusion requirement, we obtain more restrictive properties.



2.3 Definition. A convex space is unordered Baire-like if and only if it is not the union of a countable family of balanced convex rare sets.



2.4 Definition. A barrelled space is unordered quasi-Baire if and only if it is not the union of a countable family of rare linear subspaces.



The barrelled property may be cast in similar terms as follows: A convex space E is barrelled if and only if it is not the union of a sequence (nB) where B is a balanced convex rare set. For if E is barrelled and E = U nnB, then B is absorbing, and so B7 is a barrel in E. As E is barrelled, B is a neighborhood in E, and so B is not rare. Conversely, if B is a barrel in E, then E = UnnB, so that B is not rare. As B = B is balanced and convex, B is a neighborhood. Thus E is barrelled.

From inspection of the definitions above, the implications in the following diagram are clear for convex spaces. Examples in the next section establish the remaining relationships.





43


2.5 Diagram.


Baire =>unordered Baire-like.> unordered quasi-Baire





Baire-like quasi-Baire barrelled



It is not yet known whether or not the Baire and unordered Baire-like properties are distinct.



Section 2. Distinguishing examples

Amemiya and Komura [1] have shown that a barrelled and metrizable space is Baire-like. Saxon [1] has generalized this to the following.



2.6 Theorem. If a convex space E is barrelled and does not have a linear subspace topologically isomorphic to cp, then E is Baire-like.



Since cp has countably infinite dimension, cp is a union of an increasing sequence of finite dimensional subspaces each closed, since cp is Hausdorff, and so rare also. Thus cp is not a Baire space or even a quasi-Baire space. Since each balanced convex absorbing subset of cp is a neighborhood, rp is a barrelled space. Therefore cp distinguishes between barrelled and quasi-Baire spaces.





44

Each linear subspace of a convex space with the strongest convex topology is closed. Since the completion of such a space has the strongest convex topology, such a space is complete. Thus, if p is metrizable, it has a translation invariant metric d, and (p,d) is a complete metric space. By the Baire category theorem, this is a contradiction. Hence is not metrizable, and the above theorem implies that barrelled metrizable spaces are Baire-like. We shall use this in the following example which is similar to an example in Saxon [11, much simplified.

The example provides a Baire-like space which is not unordered quasi-Baire.



2.7 Example. Let R be the space of all real sequences with the product topology. Let en = (6mn )m R , and, interpreting summations coordinate-wise, let

E = [ zkc nk: (nk) strictly increasing, limk(nk k) = CO, k E R. Observe that E is a linear subspace of the convex metrizable' space R . For suppose ekakep and Y (3 e are elements 0k k

of E with k yk n their sum, where (nk) is the strictly increasing sequence formed of the elements of (klkEwU {qk kzw' For each k, let i,j be the largest integers such that p ,q are less than or equal to nk. Now nk/k max(p,q 3/(i + j) 2 axp ,q J/max~i,j} !iminp /i,q /ill and it is clear that as k increases without bound so do i and j.





45

E is not unordered quasi-Baire. For suppose g is the projection of R in the nth coordinate, then gn is a continuous linear functional which is non-zero on E. Thus N(g ) n E is a rare linear subspace of E. Clearly, E = U nN(g n) q E.

We shall now show that E is barrelled, and so, from the preceding remarks, E is Baire-like. To show that E

is barrelled, we need only show that each pointwise bounded subset A of E' is equicontinuous.

Since E is dense in R W , we may consider E' to be all of (R')' = 5' R. By density of E, A is a neighborhood in R W if and only if A n E is a neighborhood in E. Thus A is equicontinuous on R if and only if A is equicontinuous on E. Now A is equicontinuous if and only if A is bounded at each en and there is a natural number m such that
n
A~e n] =0) for all n -a m.

Assuming A is not equicontinuous, we need only find x

in E at which A is unbounded in order to show E is barrelled. If A is unbounded at some en, we are done. Suppose A is bounded at each en.
n
CLAIM: There are strictly increasing sequences (mk), (n) of natural numbers and sequences (fk) in A and (ctk) is R such that (a) f p= apen ) = k, (b) fk(em) 0 for m 2 mk,
p
and (c) nk+l > (k + 1)2, nk, Mk'

For, since A is not equicontinuous, there are f in A

and n1 with f (e ) - 0. Select a with f 1(a e 1. There
1 1 h f7' e 1 1 1.Thr






46

is M 1 with f 1(e) = 0 for m m 1. Since A is not equicontinuous, but is bounded at each e n, there are f2 in A 22
and n2 larger than 2 , n and mI such that f2(en ) 0.
1 1 2
Select a2 so that f2(aIen + a2en) = 2. An induction establishes the claim.

From (c), (nk) is strictly increasing and nk/k > k, so that x = kakenk is an element of E. Now


fk(x) = ppn ae = fk(Zp ~ ) + f (E lcre )k +0,
Ek k papen p k p=l ape n ) + k pk+ae ) = +0
kp p kp~~

from the claim and continuity of f Thus A is unbounded

at x in E.//

To show the remaining implications, excluding the first, are strict, we need an unordered quasi-Baire space which is not Baire-like. The following, suggested by S. A. Saxon, gives such an example. In addition, we shall note that the space constructed is complete.



2.8 Example. Topology aside, (tj ) is an increasing

sequence of linear subspaces of R , and so E = Un n is a linear space. Supply E with the strongest convex topology J for which each injection in: Uin-fl in) G(E,,7) is continuous. Let Sn be the closed unit ball of (t ,1 n! ). Now nSn is contained in (n+ 1)S ' and so E = U nS .
n+1 n n
Clearly E is not Baire-like if each S is rare in E. We need only show E is Hausdorff and each S is complete. For, if so, S is closed and, since Sn is balanced and con-





47

vex, Sn is rare if and only if S is not a neighborhood in E. Yet Sn is not absorbing, so S is not a neighborhood in E.

For x in E, let f k(x) be the kth term of the sequence x. Since fk is a linear functional and each fk n is continuous, fk is in E'. If x is a non-zero element of E, then some f is not zero at x. Thus E is Hausdorff.

Now suppose xm = (kn )) is in Sn and (x ) is a Cauchy net in E. Since fk is continuous, we may let in = limfk m
km n y e = limmfk xi) = )
limm and x = (ak). For each p, k akn = limm(ZPl a ) ,

by continuity of P fk in. Yet ap k n : Zkfak(m)in

Ix s 1, so 2,kP Jak n s 1. Thus Zklakn s 1, and so x is in S . Hence Sn is complete.

We note E is barrelled and the union of an increasing sequence (nS n) of balanced convex complete sets. From Valdivia [1, Th. 1], E is complete, while, from the above, E is not Baire-like.

Now E is unordered quasi-Baire. For .if E = U nE where
nn
each En is a closed linear subspace of E, then ti =Un (En flt) and E n is a closed linear subspace of (fiT). Since

the norm topology on P, is stronger than the inherited topology, each En n 1 is closed in the norm topology. Since (t l ) is a Baire space, some En n I = -1. Yet ' 1 is

dense in E since ty contains the terminating sequences which are dense in each (, I ). Thus E = E n ) C E ,
n *n 1 (n in
and so En = E. Hence E is unordered quasi-Baire.//
n






48

Finally, we remark that the barrelled requirement in definitions 2 and 4 is not redundant. For suppose B is an infinite dimensional Banach space with the weak topology a, then the weak topology for B is strictly smaller than the norm topology, and sc (B,a) is not barrelled. Yet E is not the union of a countable family of linear subspaces rare in (E,a).



Section 3. A characterization of unordered Baire-like spaces

We shall see from proposition 3.5 that each of the

four new properties is characterized by requiring a barrelled space not to be the union of translates of members of any family of sets described in its definition. A particularly useful characterization of the unordered Baire-like property has been noted by S. A. Saxon and is recorded here.



2.9 Proposition. A convex space is unordered Baire-like if and only if it is not the union of a countable family of linear subspaces each either rare or not barrelled.



Proof. Suppose E = Un F and each Fn is rare in E or not barrelled. In the former case, B = F is clearly a n n
balanced convex rare set. In the latter, suppose B is a barrel in F which is not a neighborhood in F . In this
n n
case as well, B is rare in E. For otherwise, since B
n n
is balanced and convex, B nis a neighborhood in E, and so






49

B = B n F is a neighborhood in F , a contradiction. n n n n
Finally, E = U mB , and so E is not unordered Baire-like.
mn n

Conversely, suppose E = U B and each B is balanced and n n n
convex. Let F = sp(B ), so that E = U F . Some F is
n n n n n

both barrelled and dense. Now B nn F is a barrel in F , n nn

and so a neighborhood in F . Hence B is not rare in Fn Since Fn is dense in E, Bn is not rare in E. Thus E is

unordered Baire-like.//

We note here that a non-barrelled subspace F of a convex space E is meager in a rather strong sense. For suppose B is a barrel in F which is not a neighborhood in F. Since B is balanced and convex, B is rare in F. Thus each nB is rare in E, yet F = UnfnB.



Section 4. Some permanence properties

Many of the permanence properties enjoyed by Baire spaces hold for the several related convex spaces.



2.10 Proposition. If a convex space E contains a dense linear subspace F which is (unordered) quasi-Baire [Bairelike], then E has the same property.



Proof. Suppose (B n) is a sequence of balanced convex sets, and E = U B . Now (B nl F) is a sequence of balanced
n n n
convex subsets of F, and F = Un (Bn q F) . If (Bn) is in-






50

creasing or consists of linear subspaces, the (B n F) has the corresponding properties. Now if B n F is not
n
rare in F, then, since F is dense in E, B fl F, and hence Bn, is not rare in E.//

In the context of linear mappings the concepts of almost open and open mappings have particularly simple forms. We record them in the following.



2.11 Proposition. A linear mapping f from a linear topological space E into a linear topological space F is open [almost open] if and only if for each neighborhood U in E, f[U] [f[U] r f[Ej] is a neighborhood in f[E].



Proof. One direction is immediate from the definitions. We prove the other for almost open linear mappings. Let x be in E, and U be a neighborhood of x. Now U - x is a neighborhood in E, and so f[U - xj n f[E] is a neighborhood in f[E]. The closed set f[U] - f(x) contains f[U] - f(x) = f[U - x], and so contains f[U - x] . Thus (f[U] - f(x)) nf[E] is a neighborhood in f[E], and hence f[U] n f[E] is a neighborhood of f(x) in f[E]. Thus f is almost open. The proof for open mappings is similar and simpler.//

The following is a standard context in which almost open mappings arise.



2.12 Proposition. A linear mapping from a convex space onto a barrelled space is almost open.






51

Proof. Let E be a convex space and U be a barrel which is a neighborhood in E. If f is a linear mapping of E onto a barrelled space F, then f[U] is balanced, convex and absorbing in F. Hence f[U] is a barrel in F, and so a neighborhood in F. By proposition 11, f is almost open.//



2.13 Proposition. The (unordered) quasi-Baire [Baire-like] property is preserved by continuous almost open linear mappings.



Proof. Let f be a continuous almost open linear

mapping from a convex space E, enjoying one of the above properties, onto a convex space F. F is barrelled, for if B is a barrel in F, then, by linearity, f [B] is balanced, convex, and absorbing in E. Since f is continuous, f 1[B] is closed, and so, a barrel in E. Thus f 1[B] is a neighborhood in E, and, as f is almost open, B = B = f[f1 [B]] is a neighborhood in F.

Now if F = U B where each B is balanced and convex,
n n n
then E = U nf1[B n]. By linearity, each f 1[B n] is balanced and convex. If (B ) is a sequence of linear subspaces, so
-1 -l
too is (f [B n]) Moreover, inclusions are preserved by f
-l -l
Finally, if f [Bn] is not rare, then, since f [BnI] is
-l
balanced and convex, f [B ]~ is a neighborhood in E. As
n
f is continuous, f~ [B ] is closed, and so contains f [Bn]






52

Thus f~ [B 1 is a neighborhood in E. Since f is almost
n
- -lopen, B = f[f [B 11 is a neighborhood in F, and so B
n n n
is not rare.//

The following contains an additional permanence property for convex Baire spaces.



2.14 Corollary. If an (unordered) quasi-Baire [Baire-like or convex Baire] space E is mapped onto a barrelled space F by a continuous linear mapping f, then F has the same property as E.



Proof. From proposition 12, the linear mapping f is almost open. Since f is also continuous, propositions 13 and 1.23 apply.//



2.15 Corollary. Let f be a continuous almost open linear mapping into a convex space from a convex space E containing a dense linear subspace G. If G is (unordered) quasi-Baire [Baire-like], then f[G] has the same property.



Proof. By proposition 1.22, f restricted to G is almost open. Since f is also a continuous linear mapping, the proposition gives the result.//



2.16 Corollary. If a convex space E is (unordered) quasiBaire [Baire-like], and F is a linear subspace of E, then the quotient space E/F has the same property.






53

Proof. If f is the canonical mapping from E onto E/F, then f is a continuous open linear mapping.//



Section 5. Inheritance

There has been some recent interest in inheritance of properties of convex spaces by linear subspaces of countable codimension. Evidence of this includes De Wilde and Houet [1], Levin and Saxon [1], Saxon [1,2], Saxon and Levin [1], and Valdivia [1,2]. In particular, we shall use inheritance of the barrelled property shown in Saxon and Levin [1]. This inheritance property for the Baire-like and quasi-Ba.ire properties is shown in Saxon [1,2]. For completeness, we shall record the inheritance for these two properties, and present a proof of it for the unordered

properties. For the latter, we need a fact about linear spaces given by a corollary of the following.



2.17 Proposition. If the union of two countable families S,; 2 of linear subspaces of a linear space E covers E, then one of them covers E.



Proof. Assume x,y are elements of E\Uj1, E\U 2 respectively. Since E = U(I U j2), x and y are distinct, and the line L passing through x and y is an uncountable subset of E. If an element F of 9 contains two distinct points of L, then, since F is a linear space, F contains L, and so both x and y, a contradiction. Since a U a2 is






54

countable, it covers only a countable subset of the uncountable set L. Yet L C E = U( 1 U a2), contradicting the assumption.//

This gives two corollaries.



2.18 Corollary. A linear space is not covered by a finite family of proper linear subspaces.



Proof. Suppose no linear space is covered by fewer than n proper linear subspaces, and linear space E is covered by family 3 of n proper linear subspaces. For F in 3, F / E, and so the proposition implies that the set of n - 1 proper linear subspaces 3\{F) covers E, a contradiction.//



2.19 Corollary. If a linear space E is covered by a countable family 3 of proper subspaces, then E is covered by each cofinite subfamily of 3.

Consequently, each finite dimensional subspace F of E is contained in each member of an infinite subfamily of U.



Proof. If j, is a finite subfamily of 3, then a does not cover E from corollary 18. By the proposition, j\31 covers E.

Now :7 is countably infinite, so we may index 7 uniquely as (F: n w). With its only Hausdorff linear topology, F is a Baire space, and each Fn n F is a closed subspace






55

of F. Now if F is not contained in F n then F nl F is rare in F. Since F is a Baire space and F C U F , some Fn contains F. Suppose n1 < n2 < ... < np have been chosen with F C F for 1 !: k 4 p. Since F C U F n
nk n n
there is some n > n such that F contains F. This
p+l p n+

induction defines a sequence (F ) in j of distinct elements each containing F.//



2.20 Theorem. The properties (unordered) quasi-Baire [Bairelike] are inherited by linear subspaces of countable codimension.



Proof. The ordered cases have been given elsewhere. Let E = F + sp(fxn ) where F is a linear subspace of an unordered quasi-Baire [Baire-like] space E. Some F = F + sp(fxkj) is unordered quasi-Baire [Baire-like]. Otherwise each F = U F where each linear subspace F
n m mn mn
is rare in Fn [or not barrelled]. Yet E = U F mn, so some F is dense in E [and barrelled], a contradiction. Moremn
over, F is barrelled since it is of countable codimension
n
in the barrelled space E. F is of finite codimension in Fn' Thus we need only show that the properties are inherited by linear subspaces of finite codimension. By finite induction, it is sufficient to show that the properties are inherited by linear subspaces of codimension 1. Let F be of





56

codimension 1 in E, and E = F + sp({x)). We note that F is barrelled. Let Q be a countable family of linear subspaces of F which covers F.

Assume each element of q is rare in F, and let

N = (G : G c q). No element of H contains F and M covers F, so I is not finite by corollary 18. Elements of U are of codimension at least 1, so for distinct elements H1, H2 of M, H1 n H2 is of codimension at least 2. Thus H 1n H2 + sp({x)) is a proper closed linear subspace of E, and so rare in E.

Now for y in F, the 1-dimensional subspace sp((y}) of F is contained in each of an infinite subfamily of 'a by corollary 19. Hence there are distinct elements H1, H2 of H with y contained in H1 f H2. Therefore, E = F + sp((x)) = U(H n H2 + sp((x)): H1 - H2 and H1, H2 in I).

For E either unordered quasi-Baire or unordered Bairelike, we have a contradiction. In the former case, the contradiction gives the inheritance. In the latter case, we see from proposition 17, that we may suppose each element of Q is dense in F. We need only show some element of q is barrelled. Now E = F + sp((x)) = U(G + sp((x)): G e q), and so some G + sp(tx}) is barrelled. Hence G is barrelled.//



Section 6. Productivity

Productivity for the ordered properties have been discussed in Saxon [1] and [2]. Productivity for the






57

unordered properties requires significantly different techniques from those found there. Yet these techniques also give proofs for productivity of the ordered properties. The beginning of these techniques was proposition 17. This proposition along with the following lemmas and corollaries form the basis of attack for this problem.



2.21 Lemma. If a countable family 8 of balanced convex sets of a linear space E covers E, then if B1 U 02 = (kB: k e w, B E 8), either a1 or B2 alone covers E.



Proof. Let 3k = (sp(B): B e a and k overs sp(B)), k = 1, 2. Clearly Pk covers Ujk. For B in R, (nB) is a sequence in 81 U a2, so some subsequence (n B) is in either R1 or R2. Now sp(B) = U n B, and so either R1 or R2 alone covers sp(B). Hence sp(B) is in J U a2, and this covers E. Since "k covers Ujk, either a or 9k alone covers E.//



2.22 Lemma. Let (E m) be a sequence of linear topological spaces, a a countable family of closed balanced convex subsets of the product E = 7mEm. If R covers E, then the family (kB: k c w, B e R, for some M c w, IM>ME C B) covers E. (We identify 1 EM>M as a subspace of the product mEm.)



Proof. We may suppose that 9 contains all integral multiples of its elements. Let B 1 = (B c 8: for some






58

M e W, 11m>MEm C B} and R2 = 1 We shall show that 8 covers E by showing that a2 does not cover E and applying lemma 21. If 62 is empty the result is direct. Otherwise index a2 as (Bn: n E w).

CLAIM: There are

a strictly increasing sequence (rk) of natural

numbers, and a sequence (xk) in E such that, for each k z 1,

(a) xk E Rm E and E k x B and
'2mk- I'1 k

(b) E x + y Bk for all y c 7m E


Since B is a proper subset of E, let x1 be in E\B . With m0 =1, (a) is satisfied for k = 1 and (b) is satisfied vacuously for k < 1. Suppose m < m2 < ... < mn-1' x,...,x n are chosen to satisfy (a) for k s n and (b) for k < n. If there are y. c 1miEm so that nx + y. is in B for each i, then x. + y. converges to 7 x.. Since B n 3 3 1 3 n
is closed, Ex is in Bn, a contradiction. Hence there is
n n
m > m with Znx. + y B for all y c n E and (b) n n-1 1 n M imn m
is satisfied for k < n + 1.

Now if ,nx. + E E C B then, since B is
1 i mim m n+l' n+1
n
balanced and convex, fmm E C Bn+1, a contradiction.
n
Thus there is x+1 E Fm E with En+l x B The claim
nn+1 inA 1 n+1'

follows by induction.

Since (rk) is strictly increasing, and xk E mk- EM' the coordinate-wise sums F. x. exist and are in ni E .
j2k+l j mn~mk in






59
k
From this and (b), for each k, x = kx + j 1lj j,-k+lxj BZ]k.
Hence x is not in UkOk = U"2, so 12 does not cover E.//

We have the following as a direct application of this lemma.



2.23 Corollary. Let (E m) be a sequence of linear topological spaces, and 3 a countable family of closed linear subspaces of the product E = rmEm. If 3 covers E, then the collection (F e 3: for some M, lmtMEm C F) covers E.



2.24 Lemma. Let (E M) be a sequence of linear spaces and a be a countable family of proper linear subspaces of the product E = fmEm. Suppose each F in 3 contains lm:MEm for some M E w.

If , covers E, then for some m, the collection (F E 3: E 7 F) covers E



Proof. Let Um = (F c 3: EM 7 F). Assume for each m that 3m does not cover E M. We shall obtain an element x of E not covered by 3. Note that 3 = Um m, for otherwise assume F is in 3 with E contained in F for all m. There
m
is M with myMEM C F. Now E = Em + nm>MEm C F, a contradiction, since F is a proper subset of E. For each m, let xm be in Em\J .

CLAIM: There is a sequence (am ) of scalars with


'z ,x U(31U...U3m) for all m e w.


For a = 1, a11 X 1 al Suppose a l,...,an are chosen






60

so that k a x , U(JlU...U3k) for 1 ! k s n, and let L be the line determined by Zna x and x c E \U9n if

two distinct points of L lie in an element F of lU...Un+1 then F contains L, and so F contains both Z a x. and xni, a contradiction. Since a U...Ujn+1 is countable it covers only a countable subset of the uncountable set L\(x n+). A short computation shows there is an+1 with n+1
1 a x X U(JlU... U3n+l*

Now each coordinate-wise sum Zm:_namxm is in H EmEM Let x = 7ma . For F in a, there are M and n with


1m>Mamxm E- >MEm C F c 3n' We may suppose M is larger than n, and so


amm U(JlU...UUM) so that


ImmfF E an C al U...U 3M' and so x = Mamx + ax is not in F.//
1m m >Mm m


2.25 Theorem. The properties (unordered) quasi-Baire [Baire-like] are countably productive.



Proof. Let (Em) be a sequence of convex spaces enjoying one of the properties. The product E = RmEm is barrelled. Suppose 8 is a countable family of closed






61

balanced convex sets each rare in E. If each E is
m
(unordered) quasi-Baire, then the elements of 2 are to be, in addition, linear subspaces of E.

In case 2 consists of an increasing sequence (B n)

which covers E, lemma 22 yields natural numbers k, N and M such that kBN contains H m>ME m. Thus B contains m>MEM for all n N. Hence (Bn f G)nN is an increasing sequence of balanced convex sets rare in G = l E. Since R covers E, for each m there is n N with B n E not rare in E m nm m m
Let n = max~n : 1 , m M). Hence B n G is not rare in G,
m n
a contradiction which shows E is quasi-Baire [Baire-likel.

In case the spaces (E m are unordered quasi-Baire [Baire-like], the following argument is required. If e covers E, we may suppose it contains all integral multiples of its elements. Lemma 22 shows we may assume each element B of A contains 1 E for some M. Now F = sp(B) contains
1m>M m
fl E . Moreover, since the closed balanced convex set B m>M m
is rare in barrelled space E, B is not absorbing in E, and so F is a proper linear subspace of E. Thus

(F: F = sp(B), B e S) fulfills the hypotheses of lemma 24.

This lemma gives some Em covered by (F c J: Em 7 F). In case B consists of closed subspaces of E, a = 2, and E is covered by a countable family of closed proper linear subspace (F n Em: Em 7 F e j), a contradiction of the un-ordered quasi-Baire property for E .






62

Consider F in 9 with B in a such that Em 7 F = sp(B). If F n E is dense in E , then the barrel B n E in E is not a neighborhood in E. Otherwise, since B n Em is closed in Em , B n E is a neighborhood in E . This is a m m m
contradiction since sp(B fl Em) =F fl Em / E. Hence F n Em is not barrelled. Therefore the countable family {F n Em: Em 7 F c 3) covers Em, yet no F n E is both dense in E and barrelled, a contradiction of the unordered Baire-like property for E by proposition 9.//

The following lemma is lifted out of a proof in

Saxon [1]. In its proof, we shall use twice this consequence of the bipolar theorem discussed in Horvath (1]: If B is a closed balanced convex set of a convex space E, then an element x of E is in the span of B if and only if B is bounded at x.



2.26 Lemma. If [E ) is a collection of barrelled spaces and 9 a countable family of balanced convex rare subsets of the product E = EI Ei, then there is a countable subset J of I with each B n nl E rare in Ri E .



Proof. We may suppose I is infinite, and each element of 0 is closed. Since E is barrelled, and each element B of S is a closed balanced convex set rare in E, B is not absorbing, and so E\sp(B) is not empty. Index B as (B : n e w), and let xn be in E\sp(B ).






63

Since xn is not in sp(B n), there is a sequence (f mn)m in B C E' = E' which is unbounded at x . For each
n Ii n
f there is a finite subset I of I with f [l E.]= (0). inn nn inn I\I 1

Let J = U I n, and F =lE. We shall show each B n F

is rare in F.

Define (y ) in F by y n n, where 7 is the projection of E onto F. By choice of J fiy = ffX for all mn, and so (f ) is unbounded at y Now (f ) C B ,
mn m n mn m n
and B D B n F so (f ) C B C (B n F) . But B n F
n n ' mn m n n n
is a closed balanced convex set of E with (Bn n F)0 = (Bn n F) unbounded at y n. Therefore yn is not in the span of Bnl Fn . Since yn is in F, Bn n F is not absorbing in F, and so not a neighborhood in F. Yet Bn n F is closed balanced and convex in F, and so Bn n F is rare in F.//
n


2.27 Theorem. The properties (unordered) quasi-Baire [Baire-like] are productive.



Proof. We use the notation of the previous lemma. If R covers E, then (B f F: B E 8) covers F. If the cover R contradicts one of the properties for E, then (B f F: B e 2) does so for the countable product F = 7 E . Since the properties are preserved by countable products,

the proof is complete.//














CHAPTER 3

Category Analogs

In this chapter, we discuss concepts in convex spaces analogous to category in linear Baire spaces. Since the analogs to the Baire property discussed in the previous chapter entail linear space concepts such as absolute convexity, the analogs to category are initially problematic. The success of the definitions adopted here is reflected in the analogs found for some standard category theorems in linear topological spaces. In particular, proposition 5 is analogous to the characterization of linear Baire spaces in proposition 1.26.



Section 1. Analogs of meagerness

We simplify the presentation by restricting the discussion to the context of the Baire-like property. Only slight changes in wording are necessary for the other properties, and, in the cases of the unordered properties, the proofs are generally shortened.



3.1 Definition. A subset A of a convex space E is meagerlike in a subset B of E if and only if there are an increasing sequence (Bn) of balanced convex subsets of E and a sequence


64






65

(x ) in E such that A is contained in U (x + B ) and each
n n n n
x + B is rare in B. A is said to be meager-like if A is
nn
meager-like in E.



Similar definitions may easily be formed using

[unordered] quasi-meager and unordered meager-like, and, with minor changes, for example, the insertion of the barrelled property, the following theorems apply to all of these properties.

The theorems stated in this and the next section are analogs of standard category theorems for linear topological spaces which may be found in Kelley and Namioka [1]. These theorems may be obtained by replacing "Baire-like" by "Baire," "meager-like" by "meager," and "convex space" by "linear topological space." Since there are Baire-like and unordered quasi-Baire spaces which are not Baire spaces, these theorems extend category-like structures to some spaces about which the standard category theorems provide

no information.



3.2 Proposition. Translates and scalar multiples of meagerlike subsets of a convex space are meager-like.



Proof. Suppose A C U (x + B ) and each x + B is
n n n n n rare in E. Thus y + a(x + B ) is rare in E, for y an n n
element of E and a a scalar. But y+cxAC U ((y+ax) +aB n).//

The proof of the following is clear.






66

3.3 Proposition. If a set A is meager-like in a subset B of a convex space, then each subset of A is meager-like in B.



For the [unordered] quasi-Baire case we require E to be barrelled in the following propositions.



3.4 Proposition. If a linear subspace F of a convex space E contains a subset which is not meager-like in E, then F is Baire-like.



Proof. Suppose F = U B with each B a balanced conn n n
vex rare subset of F. Each Bn is rare in E therefore F is meager-like in E, and each subset of F is meager-like in E.//

For descriptive convenience, we shall call a translate of a linear subspace a flat set, and note that a flat set contains all the points of a line if it contains two distinct points of the line.



3.5 Proposition. A convex space is Baire-like if and only if it is not meager-like.



Proof. Suppose E is a convex space. If E is not meagerlike, then clearly, E is Baire-like. Conversely, suppose E = U (x + B ) where each B is balanced and convex. Now
n n n n






67

E = U mnmB , for otherwise, suppose x is in E but not in U sp(B ) = U mTB n If sp(x) meets x + B at two disn n n n
tinct points, then the flat set xn + sp(B ) contains sp(x) and, therefore, contains 0, and so is the linear subspace sp(B n). Thus x is in xn + sp(B n) = sp(B ), a contradiction. Therefore sp(x) meets xn + B in at most one point, and so the uncountable set sp(x) meets U (x + B ) = E in a countable set, which contradicts the n n n
choice of x.

Now, with (B n) increasing, each mB is contained in kBk, for k larger than m and n. Thus E = UmnmBn = UkkBk. Since E is Baire-like, and (nB ) is an increasing sequence
n
of balanced convex sets, some nBn is not rare. Hence x + B is not rare, and so E is not meager-like.// n n


3.6 Proposition. If a subset A of a Baire-like space is absorbing at some point, then A is not meager-like.



Proof. By proposition 2, we may assume A is absorbing. By proposition 3, we need only show that the balanced absorbing core of A is not meager-like. Thus we may assume A is balanced and absorbing.

Supposing A C U (x n + B ), each B is balanced and n(nx n
convex, and x is in A\UmnmBn = A\Unsp(Bn). Just as in the previous proof, we may show that sp(x) meets Un (xn + Bn) in a countable set. Hence we may choose |al s 1 so that






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ax is not in U (x + B ). Since A is balanced, ax is in
n n n
A C U (x + B ), a contradiction. Thus A C U sp(B ).
n n n n n
Now A absorbs each point of E, so each point of E is in some sp(B ). Thus E = U sp(B ) = U mB = U kB . Since
n n n mn n k k
E is Baire-like, some kBk is not rare, and so xk + Bk is not rare. Therefore A is not meager-like.//

For the unordered cases, a shorter proof follows from the observation that a countable union of unordered meagerlike [quasi-meager] sets is unordered meager-like [quasimeager].



3.7 Corollary. Each non-empty open subset of a Baire-like space is not meager-like.



Proof. Each non-empty open subset of a convex space is absorbing at some point.//



Section 2. The subgroup theorem and applications



3.8 The difference theorem. If a subset A of a convex space E is not meager-like, and some open set U has a meager-like symmetric difference, U A A, with A, then A - A is a neighborhood in E.



Proof. The open set U is non-empty, for otherwise,

A = (A\U) U (U\A) = U A A, and so A is meager-like, a contradiction. Proposition 4 shows that E is Baire-like.






69

We note that if D is meager-like, then D U (x + D)

is also. For suppose D C Un(xn + Bn), define C2n C 2n-= B n, and y2n x y2n-1 x + xn, so that D U (x + D) C Un n + B ) U [x + Un n + B )] =


= Un (2n + C2n) U [Un(x+ xn+ C2n-1)]= Un(Yn+ Cn)' and (C n) is increasing when (Bn ) is.

Now [U n (x + U)]\[A n (x + A)] =

[U q (x+U)\A] U [U r (x+U)\(x+A)]C (U\A) U [(x+U) \(x+A)]

(U\A) U [x + (U\A)] C (U A A) U (x + (U A A)).

With D = U A A above, we see that the last set is meager-like. Therefore, by proposition 3, [U n (x + U)]\[A n (x + A)] is meager-like.

If U meets x + U, then U n (x + U) is not meager-like by corollary 7, hence A must also meet x + A. Thus U - U = (x e E: U n (x + U) 0) is contained in [x E E: An (x+A) 0)= A - A. Since U is a non-empty open set, U - U is a neighborhood, and so too is A - A.//



3.9 The subgroup theorem. If an additive subgroup G of a convex space E has a subset A which is not meager-like in E, and some open set U has a meager-like symmetric difference, U A A, with A, then G = E.

Thus, if G is any proper subgroup of E, then G is meager-like or for no open set U is U A G meager-like.






70

Proof. By the difference theorem U = (A - A) is a neighborhood. Since G is an additive group, G D G - G D A - A D U. Thus G D U nU = E.//
n


3.10 Corollary. If a subset A of a Baire-:ike space E is absorbing at some point, and some open set U has a meagerlike symmetric difference, U A A, with A, then A - A is a neighborhood.



Proof. By proposition 6, A is not meager-like, so the difference theorem applies.//



3.11 Corollary. Let f be an additive mapping from a convex space E into a convex space F. If, for each neighborhood V in F, there is a subset A of f 1[V] and an open set W of E such that A is not meager-like and W A A is meager-like,

then f is continuous.

Dually, if, for each neighborhood U in E, there is a subset A of f[U] and an open set W of F such that A is not meager-like and W A A is meager-like, then f is an open mapping onto F.



Proof. For a neighborhood V' in F there is a neighborhood V with V - V C V'. For some subset A of f [V] and some open set W of E, A is not meager-like and W A A is meager-like. By the difference theorem, A - A is a neighborhood in E.






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Now, by additivity of f, we have A - A C f 1[V] - f1 [V]= f 1[V - V] C f I[V']. Thus f 1[V'] is a neighborhood in E, and so f is continuous.

Dually, the additivity of f gives f[E] - f[E] =

f[E - E] = f[E), so f{E] is an additive subgroup of F. Since E is a neighborhood in E, there is a subset A of f[E] and an open set W of F with A not meager-like in F and A A W meagerlike. Thus f[E] = F by the subgroup theorem.

For a neighborhood U' in E, there is a neighborhood U in E with U - U C U'. For some subset A of f[U] and some open set W of F, A is not meager-like and W A A is meagerlike. By the difference theorem A - A is a neighborhood in F. Now, by additivity of f, we have A - A C f[U] - fUl = f[U - U] C f[U'I. Thus f[U'] is a neighborhood in F, and so f is an open mapping onto F.//



3.12 corollary. Let f be an additive mapping fram a convex space E into a convex space F. If E is Baire-like, then f is continuous if and only if for each open neighborhood V in F, f 1[V] is absorbing and for some open set W of E, W A f 1[V] is meager-like.

Dually, if F is Baire-like, then f is an open mapping

onto F if and only if for each open neighborhood U in E, f[U] is absorbing and for some open set W of F, W A f[U] is meager-like.






72

Proof. Let V be an open neighborhood in F. If f

is continuous, then f 1[V] is an open neighborhood in E, and so is absorbing. With W = f~ [VI, W A f 1[VI = j is meager-like.

Conversely, f- 1[VI is absorbing in the Baire-like

space E, so by property 6, f 1[V] is not meager-like. For some open set W of E, W A f [V] is meager-like. With A = f 1[V], corollary 11 applies.

Dually, let U be an open neighborhood in E. If f is an open mapping f[U] is an open neighborhood in F, and so is absorbing. Moreover, with W = f[U], the set WA f[U]=9 is meager-like.

Conversely, f[U] is absorbing in the Baire-like space F, and so is not meager-like. Since W A f[U] is meagerlike for some open set W of F, corollary 11 applies.//



3.13 Corollary. Let f be an additive mapping from a convex space E into a convex space F. If f[A] is bounded for some subset A of E such that A is not meager-like and W A A is meager-like for some open set W of E, then f is continuous.

Dually, if for some bounded set A of E, f[A] is not meager-like and W A f[A] is meager-like for some open set W of F,then f is an open mapping onto F.



Proof. Let V be a neighborhood in F. For some integer n, f[A] C nV. By additivity of f, A C f 1[nV] = nf I[VI.






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Thus f~ [V] contains A/n which is not meager-like, yet for some open set W of E, W A A and so (W/n) A (A/n) is meager-like. Thus corollary 11 applies, and f is a continuous mapping.

Dually, suppose U is a neighborhood in E. For some integer n, A C nU, and so by additivity of f, f[A] C f[nU] = nf[U]. Thus f[Aj/n C f[U]. Now f[A]/n is not meager-like, yet for some open set W of F, W A f[A] and so (W/n) A (f(A]/n) is meager-like. Hence corollary 11 applies, and f is an open mapping onto F.//



Section 3. Analogs of the condition of Baire

A set A of a topological space X is said to satisfy the condition of Baire if there is an open set U whose symmetric difference, U A A, with A is meager. Such sets are discussed in Kelley and Namioka [1], and the analog of these sets has been used repeatedly in the previous section. The collection G of all subsets of X satisfying the condition of Baire contains X and is closed under countable unions and differences, that is, G is a a-algebra. If U is open in X, then U A U = P is meager, and so U is in G. Hence G contains all open sets, and so, in particular, each Borel set satisfies the condition of Baire. The analogous results do not hold even for unordered meager-like.

2
3.14 Example. Let A be the unit circle in E = R . A is closed and so is a Borel set, yet for no open set U is






74


U A A unordered meager-like. For, if U is a non-empty open set, the U A A contains the non-empty open set U\A which is non-meager. For U A A to be unordered meagerlike, U A A must be meager, and so U must be empty. However, no countable collection of line segments covers A since each line segment meets A in at most two points and A is uncountable.//

The class analogous to G for unordered meager-like or quasi-meager is, however, closed under countable unions and finite intersections and contains all open sets.

In the following, unordered meager-like may be replaced by unordered quasi-meager.



3.15 Proposition. Let G be the collection of all subsets A of a convex space with U A A unordered meager-like for some open set U. G contains all open sets and is closed under countable unions and finite intersections.



Proof. For each open set U, U A U = 0 is unordered

meager-like, and so U is in G. Suppose (A n) is a sequence in G and (U n) is a sequence of open sets with each Un A An unordered meager-like. Let A = U nA and U = U nU . Now UAA= (U mUm\U n A n) U (Un A n\UmUM) C [Um (U \A )] U [U n(A \U n)] C UMI(Um\A ) U (A \UM m(Um A Am), which is unordered
meager-like. Thus Un An is in G.

Also (U1 n U2) A (A1 nA2) = [(U1 nU2)\(A 1 A2)IU[(A1fnA2)\(Ul fU2)I C [(U \A1) U (U2 \A2)] U [(A1 \U 1) U (A2\U2)] = (U A A 1) U (U2 AA2),






75

which is unordered meager-like. Thus A n A2 is in U, and by induction G is closed under finite intersections.//



Section 4. An open question

An important application of the Banach subgroup theorem provides the following: If (fn) is a sequence of continuous linear mappings of a linear topological space E into a pseudo-metrizable linear topological space, then the set of points x for which (f n(x)) is Cauchy is either meager or identical with E. The proof of this result uses the closure under countable intersections of the family of sets satisfying the condition of Baire. An analog of the result is not known to be true.














CHAPTER 4

Applications

In this final chapter, we shall consider a number of applications of theorems and concepts developed above. In particular, we shall consider refinements of a category result on continuous linear images of Ptak spaces as well as a modification of the Robertson and Robertson [1] closed graph and open mapping theorems.



Section 1. Initial open mapping and closed graph theorems

In the following, we shall depend heavily on properties of Pta'k spaces discussed in Horvdth [1].



4.1 Definition. A convex Hausdorff space F is a Ptk space if and only if each continuous linear mapping from F into a convex Hausdorff space is open if it is almost open.



Examples of Ptdk spaces are any Frdchet space and the

dual of any reflexive Frechet space under the strong topology.

We shall use the following closed graph and open mapping theorems proved in Horva'th [1]. We record the first without proof. The second follows from proposition 2.12 and the above definition.


76






77

4.2 Theorem [Robertson and Robertson]. If f is a linear mapping from a barrelled Hausdorff space E into a Ptdk space F which is closed in E x F, then f is continuous.



4.3 Theorem [Ptdk]. If g is a continuous linear mapping from a Ptdk space F onto a barrelled Hausdorff space E, then g is an open mapping.



Section 2. Extensions of a theorem of Banach

Banach [1, p. 38] showed that the continuous linear image of a complete metrizable linear space in a space of the same kind is either meager or the whole space. Robertson and Robertson [1] extended this result in convex spaces. We refine their results by specifying in what sense the image is meager.



4.4 Proposition. If g is a continuous linear mapping from a Ptak space F into a convex Hausdorff space E, then g is a surjection if g[E] is both dense and barrelled.



Proof. Since E is Hausdorff, we need only show that

g[F] is complete in order to show g[F] is closed. From this, g[F] = g[F] = E.

Since g[F] is barrelled and Hausdorff, g is an open mapping from theorem 3. Let r be the canonical mapping from F onto the quotient space H = F/N(g), and g be the






78

one-to-one linear mapping from H into E induced by g, so that g = g o 7. Since g is open and continuous, g is also. Therefore g[F] = g[H] is the embedding of H in E by g.

Now E is Hausdorff and g is continuous, so N(g) is a closed linear subspace of the Ptdk space F. Thus H = F/N(g) is a Ptik space, so g(F] is as well. Yet a Ptdk space is complete, thus g[F] is complete.//

Clearly a non-dense linear subspace is meager. The remark following proposition 2.9 shows that a linear subspace which is not barrelled is meager. Hence the above proposition includes Banach's result for Frdchet spaces.

As an application of the proposition, we have the following.



4.5 Proposition. Let v n:Fn - F) be a countable family of linear mappings vn from the Ptik spaces F into the linear space F which is covered by (vn [Fnfln'

If F is given a convex Hausdorff topology for which

each vn is continuous, then some v is a surjection or F is not unordered Baire-like.



Proof. Suppose each v [F ] n F. From proposition 4, each v n[F n] is rare or not barrelled. Yet F is covered by {vn [F ni)n and so F is not unordered Baire-like by proposition 2.9.//






79

An example of S. A. Saxon shows the above statement to be false when "unordered" is omitted.*

This proposition may be applied to the space E described in example 2.8 to show immediately that E is not unordered Baire-like.

The following is a further refinement of a category result of Robertson and Robertson [1].



4.6 Proposition. Let [v :F - F) be a countable family of n n ncuj
linear mappings vn from the Ptdk spaces Fn into the linear space F which is covered by [vn[F n3). Let F have a convex topology for which each v n is continuous.

If g is a continuous linear mapping from F into a convex Hausdorff space E, then g is a surjection or g[F] is unordered meager-like.



Proof. Suppose g[F] is not unordered meager-like.

Clearly, g[F] is dense. By the unordered version of proposition 3.4, g[F] is unordered Baire-like. Now g(F] = U ng[v [F ]], and so some g[v [F ]] is dense in g[F] and barrelled by proposition 2.9. Since g[F] is dense in E, g[vn [F ]] is dense in E. Proposition 4 applies for the continuous linear mapping g o v . Hence g o v n, and so g, is a surjection.//


* S. A. Saxon, personal communication.





80

Section 3. The Robertson and Robertson theorems

We now state and prove the Robertson and Robertson

closed graph theorem with convex Baire replaced by unordered

Baire-like. The essential modification in the proof occurs in the second line, the rest of the proof is explicated for

completeness.



4.7 Theorem [Robertson and Robertson]. Let a family

(u.:E. - El. and a countable family {v :F - F) contain

the linear mappings u and v from the Hausdorff unordered Baire-like spaces E. and the Ptak spaces F into the linear
1 n
spaces E and F respectively. Suppose E has the strongest convex topology for which each u. is continuous, and F has

a convex Hausdorff topology for which each v is continuous.

If F is covered by (v [F ]) , then each linear mapping

f from E into F which is closed in E x F is continuous.



Proof. Suppose E is a Hausdorff unordered Baire-like

space. Now E = f [F] = Un f [v n[F ]], so some L= f 1[v [F n]]

is dense in E and barrelled by proposition 2.9. We obtain

the following diagram


fL IL V = Vn

f [v [F ]] L F F
n n
v T1

--1 ~M = F /N (v)
g = V 0 f L n





81

where 7 is the canonical mapping from F onto the quotient space M, and v is the continuous linear mapping from M into F induced by v = vn so that v = vo .

Since v is one-to-one and onto v[F n = v[M] D f[L], we have g = v 0 fL a well-defined linear mapping from L into M. Let h be the continuous mapping from L x M to E x F defined by h(x,y) = (x,v(y)). Identifying f and g with their graphs, we have g = h1 [f] and, since f is closed in E x F, g is closed in L x M.

Now F is Hausdorff, so N(v) is a closed linear subspace of the Ptak space F. Therefore M = F n/N(v) is a Ptak space. Also L is a barrelled Hausdorff space. By theorem 2, g is continuous.

A Ptak space is complete and Hausdorff, and so, as L

is dense in E, g has a unique continuous extension Q from E into the Ptak space M. For x in L, v(g(x)) =v(v 0 f(x)) = f(x). If this holds for all x in E, then f = vo c, and so f is a composition of continuous functions.

Assume x' is in E with f(x') v(g(x')). Since f is closed in E x F, there are neighborhoods U in E and V in F with (x' + U) x (v(g(x')) + V) disjoint from f. By continuity of vo g, there is a neighborhood U' in E with U'CU and Vo g[U'J C V. Since L is dense in E, there is an element x in L n (x' + U'). Now f contains (x,f(x)), and (x,f(x)) = (x,v o g (x)) E (x' + U') x (Vo 0 [x' + U']) C (x' + U) x (va g (x') + V), a contradiction. Thus vo g (x) = f(x) for all x in E.





82

For the general case, we need to show that each mapping f. = fo u from E into F is continuous. Fix i and let h be the continuous mapping from E. X F into E X F defined by h(x,y) = (u (x),y). Clearly, f. = h1 [f and, since f is closed in E x F, the mapping f. is closed in E. X F. The initial portion of the proof shows that f. is continuous.//

There is the corresponding open mapping theorem which we record without proof.


4.8 Theorem. Let the hypotheses of theorem 7 hold. If F is covered by (v n [Fn n, then each continuous linear mapping g from F onto E is an open mapping.



Valdivia [2] has studied a closed graph theorem which includes the Robertson and Robertson closed graph theorem. In place of Ptdk spaces, he uses a type of space characterized by a closed graph property which is satisfied by Ptak spaces. In place of convex Baire spaces, Valdivia defines and uses infra-Baire spaces.



4.9 Definition. A convex space E is an infra-Baire space if and only if there is a convex Baire space G such that E is embedded in G as a linear subspace of finite codimension.



Since a convex Baire space has the unordered Bairelike property, which is inherited by linear subspaces of





83

countable codimension, an infra-Baire space is unordered Baire-like. Clearly, a space which distinguishes between infra-Baire and the convex Baire properties answers in the negative the question of inheritance of the Baire property by linear subspaces of countable codimension.

Although Valdivia (2] proves several permanence properties for infra-Baire spaces, which are shared by unordered Baire-like spaces, productivity of infra-Baire spaces is left an open question. It may be seen that the infra-Baire property is inherited by linear subspaces of countable codimension. The productivity and inheritance properties for unordered Baire-like spaces shows that unordered Baire-like in theorems 7 and 8 may be replaced by products of countable-codimensional subspaces of convex Baire spaces or, equivalently, by products of infra-Baire spaces.














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[1] Espaces Vectoriels Topologigues, Chapitres
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[1] Introduction to General Topology, D. C. Heath,
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[1] Normed Linear Spaces, Ergebnesse der Mathematik
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84








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BIOGRAPHICAL SKETCH


Aaron Rodwell Todd was born December 25, 1942, in El Portal, Florida to Caroline Osborne Hall and Edmund Neville Todd. After graduating from Miami Edison Senior High School in 1960 and the University of Michigan in 1964, he taught mathematics for three years in Ghana.

Janet Margaret Dakin and he were married December 21, 1966 in Llanrhaerdr, Wales. They both attended the University of Leeds for a year beginning October, 1967. Julian Garfield Todd was born to them October 2, 1968, in Sutton Coldfield, England.

All three arrived January, 1969, in Gainesville, Florida where Julian enjoyed his most formative years while his parents taught mathematics and English literature and attended the University of Florida.


87










I certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Dcotor of Philosophy.





J.'K. )r'boks, Chairman
Professor of Mathematics



I certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philosophy.





S. A. Saxon
Assistant Professor of Mathematics



I certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philosophy.





Z. Pop-Stojanovic /
Professor of Mathematics



I certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philosophy.





A.K. Varma
Associate Professor of Mathematics









I certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philosophy.





A. R. Bednarek
Professor of Mathematics





I certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philosophy.





Ward Hellstrom
Professor of English





This dissertation was submitted to the Department of Mathematics in the College of Arts and Sciences, and to the Graduate Council, and was accepted as partial fulfillment of the requirements for the degree of Doctor of Philosophy.

August, 1972


Dean Graduate School




Full Text

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Linear Baire Spaces and Analogs of Convex Baire Spaces By Aaron R. Todd A DISSERTATION PRESENTED TO THE GRADUATE COUNCIL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY UNIVERSITY OF FLORIDA 1972

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To the Heroic Vietnamese Peopl

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ACKNOWLEDGEMENTS The author wishes to acknowledge the debts owed the members of his committee, particularly his chairman. Dr. J. K. Brooks, who, perhaps without realizing it, gave much needed encouragement to the author at several strategic moments in his research. A special debt is owed by the author to his dissertation supervisor. Dr. S. A. Saxon, who suggested the work undertaken and whose criticisms helped prevent mathematical blunders and understatements of the author from reaching print. Although not directly involved with this dissertation except for several concentrated hours of patient attention to an exposition of some of the author's discoveries. Dr. P. Bacon is gratefully recognized for a major exercise of the author's critical appreciation of mathematics through a year-long course in algebraic topology presented in the Texas style. Lastly, the author notes the financial support for his graduate studies obtained through several pleasurable teaching appointments in the Department of Mathematics of the University of Florida, a University of Florida Graduate School Fellowship 1969 70 , a National Science Foundation Summer Traineeship 1971 and the economies of his family. iii

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TABLE OF CONTENTS Acknowledgements Abstract ' Introduction Section 1. Overview Section 2. Some conventions j definitions and observations Chapter 1. Properties of Baire Spaces Section 1. Rare and meager sets Section 2. Baire spaces Section 3. Almost open and open mappings Section 4. Some permanence properties of Baire spaces Section 5 . A Baire category theorem Section 6 . Pseudo-completeness of Oxtoby Section 7. Productivity of the pseudocomplete property Section 8. Pseudo-completeness in linear topological spaces Section 9. A productive class of convex Baire spaces Section 10. Inheritance of the Baire property Page iii vi 1 2 4 8 8 14 16 18 21 23 26 28 32 33 iv

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41 Chapter 2. Convex Baire Space Analogs Section 1. The new convex spaces 41 Section 2. Distinguishing examples 43 Section 3. A characterization of unordered Baire-like spaces 48 Section 4. Some permanence properties 49 Section 5. Inheritance 53 Section 6 . Productivity 56 Chapter 3. Category Analogs 64 Section 1. Analogs of meagerness 64 Section 2. The subgroup theorem and applications 68 Section 3. Analogs of the condition of Baire 73 Section 4. An open question 75 Chapter 4. Applications 76 Section 1. Initial open mapping and closed graph theorems 76 Section 2. Extensions of a theorem of Banach 77 Section 3. The Robertson and Robertson theorems 80 V

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Abstract of Dissertation Presented to the Graduate Council of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy LINEAR BAIRE SPACES AND ANALOGS OF CONVEX BAIRE SPACES By Aaron R. Todd August, 1972 Chairman; J. K. Brooks Major Department: Department of Mathematics A locally convex linear topological space, which is barrelled, is a Baire space if and only if it is not the union of an increasing sequence of rare sets. By replacing "sets" with "linear subspaces" or "balanced convex sets" and by leaving in or removing "increasing," we obtain four distinct classes of convex spaces which fall between the convex Baire spaces and the barrelled spaces, and enjoy the following two permanence properties in addition to those known for linear Baire spaces: Each class (1) is closed under arbitrary products, and (2) contains the linear subspaces of countable codimension of each of its elements. A contribution of this dissertation is a technique which gives the productivity property, (1) , for all four classes of spaces. Also recorded are proofs of the inheritance property, (2), for these classes. Partial results concerning these two permanence properties for the linear Baire spaces are given. vr

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In particular, productivity is established for certain classes of linear Baire spaces, and connections are discussed of the inheritance property with a long-standing and unresolved question posed by V. Klee and A. Wilansky about the null space of a linear functional in Banach space theory. Also considered are analogs in the new spaces of sets of first and second categories which allow formulation of several theorems analogous to standard category theorems. These theorems give category-like structure in spaces which need not be Baire spaces. In particular, we obtain an analog in each of the new spaces of the subgroup theorem of S. Banach. Applications of these concepts include refinements and generalizations of a category theorem of S. Banach on the continuous linear image of a Frdchet space as well as a simple reformulation of the closed graph and open mapping theorems of A. P. Robertson and W. J. Robertson. In particular, this reformulation allows the replacement of convex Baire spaces in the original theorems by products of countable-codimensional linear subspaces of convex Baire spaces. VLl

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INTRODUCTION Basic to the theory of Banach spaces are the closed graph theorem and the principle of uniform boundedness. Natural to and characterized by the latter in its general formulation for locally convex topological vector spaces are the barrelled spaces. Central to the more general formulations of the closed graph theorem found in Day [1] and Robertson and Robertson [1] are the linear Baire spaces. Additionally, each locally convex Baire space is a barrelled space, and so it is natural to investigate the permanence properties of these two types of spaces as well as those of intermediate types of spaces. The original aim of the research recorded here was to resolve the following two questions; (1) Is the product of each family of linear Baire spaces a linear Baire space? and (2) Is each linear subspace of countable codimension in a linear Baire space also a linear Baire space? For "barrelled" in place of "linear Baire", the affirmative answer to the first question is well known, and the affirmative answer to the second question was established independently and through different methods by Saxon and Levin [1] and by Valdivia [1] . These observations suggest the additional aims, adopted here, of investigating the questions for several new types of convex spaces which fall 1

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2 between the locally convex Baire spaces and the barrelled spaces. In following this course, the author is deeply indebted to his supervisor, Dr. S. A. Saxon, who suggested the basic definitions, and some of whose joint and independent work is necessarily discussed here. Section 1. Overview We generally follow the terminology and notation of Horvath [1] ; however we adopt some conventions of Robertson and Robertson [2] and of Kelley and Namioka [1] . In particular, we use "convex space" in place of "locally convex topological vector space" and "linear space" in place of "vector space" . In the chapter on analogs of convex Baire spaces, we shall find that the four new convex spaces enjoy several permanence properties in addition to those known for convex Baire spaces. In particular, a contribution of this dissertation is a technique which shows all the new properties . are productive. In the chapter on applications, it is shown, with a minor change in the proofs found in Horvath [1] , that one of the four new properties, unordered Baire-like, may be used in place of the convex Baire property in the closed graph and open mapping theorems of Robertson and Robertson [1] . Although the four new properties are pairwise distinct, it is yet to be shown that the convex Baire property and unordered Baire-like property are distinct. However,

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3 the affirmative answers obtained in the second chapter to questions ( 1 ) and ( 2 ) above for the unordered Baire-like property together with the new formulation of the Robertson and Robertson theorems implies that convex Baire spaces may be replaced in the original theorems by products of countable-codimensional linear subspaces of convex Baire spaces. We also find in this chapter that the unordered Baire-like property includes that infra-Baire property defined and used by Valdivia [ 2 ] in a generalization of the Robertson and Robertson closed graph theorem. The chapter on Baire spaces is primarily introductory. However, in the latter half, we discuss, for linear topological spaces, the concept of pseudo-completeness introduced in general topology by Oxtoby [ 1 ] . This has bearing on the question of productivity of the Baire property for a class of linear Baire spaces. Another result of Oxtoby [ 1 ] easily implies that products of separable, pseudo-metrizable convex Baire spaces are convex Baire spaces. Also considered in i±iis chapter is a partial result for the question of inheritance of the Baire property by linear subspaces of countable codimension in a linear Baire space. In the chapter on category analogs, we demonstrate some success for concepts in the new spaces analogous to concepts of first and second category. The success is indicated by the extension to the new spaces of the Banach subgroup theorem as well as several other standard category theorems presented in Kelley and Namioka [ 1 ] .

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4 Section 2. Some conventions, definitions and observ ations The reader will notice some ellipses in notation and uses of alternate notations, which, once forewarned, the reader may find contribute to clarity and simplicity of expression. For example, ^I^i used in place of and n E. respectively, where uo is the set of positive integers iel i and I is an arbitrary index set. Let E be a linear topological space. Thus E is a linear space whose scalar field is either the real or the complex numbers and which has a topology for which vector addition and scalar multiplication are continuous. E* represents the vector space of linear functionals on E, while E' is the linear subspace of E* consisting of the continuous linear functionals on E. If B is a subset of E, we say the subset of E', {f € E’:lf(x)l s 1 for all X in b} , is the polar of B and represent it by B°. Similarly, if C is a subset of E', the polar C° of C is the subset of E, {x e E:|f(x)l €, 1 for all f in C] . For an element f of E*, we let N(f) represent the null space f“^[{0}] of f. N(f) is closed in E if and only if f is continuous. Suppose B is a subset of E. We say B absorbs a subset A of E if, for some positive real number p> , B contains oA for all la| ^ B. We say B is absorbing if B absorbs each singleton of E. We say B is absorbing at an element x of E if B X is absorbing. We say B is balanced if it contains aB for all \a\ ^1. We say B is convex if it contains [ax + (1 a)y:0 ^ a ^ 1, x and y in b] . We say A is

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5 absolutely convex if it contains {ax + Py:lal + 1 /3 1 ^ 1, X and y in b} . The closure of a set with any of the above properties retains that property. A set is balanced and convex if and only if it is absolutely convex. The balanced absorbing core of a set A is the largest balanced absorbing subset of A. An absorbing set has a non-empty balanced absorbing core. The set of all finite linear combinations of elements of a set A is called the span of A and is represented by sp (A) ; it ts the smallest linear subspace containing the set A. A balanced convex set B is absorbing if and only if, for each x in E, there is some positive integer n for which nB contains x. Thus for a balanced convex set B, we have sp(B) = Uj^nB. We say B is a barrel if it is absorbing, balanced, convex and closed in E. If U is a neighborhood of 0 in E, we shall say that U is a neighborhood in E or, simply, U is a neighborhood. A subset A of the dual E' of a convex space E is equicontinuous if and only if A° is a neighborhood. A linear topological space has a local neighborhood base at 0 of absorbing balanced closed sets; a convex space has one of barrels. A convex space has its strongest locally convex topology if and only if each absorbing balanced convex set is a neighborhood. A convex space in which each barrel is a neighborhood is said to be barrelled . A convex space E is barrelled if and only if each pointwise bounded subset of its dual E'

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6 is equicontinuous . Each convex space with its strongest locally convex topology is barrelled. is the convex metric space of all real sequences under term-wise convergence^ or ^ equivalently, the product of a countably infinite family of copies of the real numbers R. For 1 ^ p < is the Banach space of complex sequences ^ 1 /id X = (a^) with the norm HxH^ = cp is the convex space of infinite countable Hamel dimension with its strongest locally convex topology. cp is topologically isomorphic to the direct sum given the strongest locally convex topology for which each injection is continuous. cp is barrelled. Suppose B is a balanced convex closed subset of a linear topological space E. B has a non-empty interior if and only if B is a neighborhood. If, in addition, E is a barrelled space, then B has a non-empty interior if and only if B is absorbing . If O' is a topology for a space X, we may write (X,IT) ; if d is a pseudo-metric on X, we may write (X,d) . We represent the closure of a set A by A , its interior by A . Suppose (x. ) . is a net on a linear topological space E. (x.)i^I is said to be Cauchy if the net (x^ x ^ converges to 0 in the linear topology on E. Suppose d is a pseudo-metric on the set E, and (Yj^) ^ sequence in E. (y^) is said to be Cauchy in (E,d) or d-Cauchy if the net Mfv V )) X of real numbers converges to zero. E is v^v^m^ -^n (m,n) eujxua

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7 said to be complete if each Cauchy net converges in the linear topology of E to an element of E. (E,d) is said to be complete or, equivalently, E is said to be d-complete if each d-Cauchy sequence converges in the pseudo-metric topology of E to an element of E. If d induces the linear topology of E, and d is translation invariant, then E is complete if and only if E is d-complete. A linear topological space is a pseudo-metrizable space if and only if it has a countable local neighborhood base at 0. A linear topological space which is pseudometrizable has a translation invariant pseudo-metric. A convex metrizable space which is complete is said to be a Frdchet space . In particular, is a Frdchet space.

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CHAPTER 1 Properties of Baire Spaces In this chapter we define and investigate properties of rare sets, meager sets, and Baire spaces. The most important and best known of these properties is the Baire category theorem for complete metric spaces. A scheme of Oxtoby [1] is given which covers this as well as other Baire category theorems. This scheme has a bearing on productivity of a class of Baire spaces. Simplification of these properties and theorems in the setting of linear topological spaces are given, and the chapter ends with a discussion of an open question about inheritance of the Baire property in linear topological spaces. Section 1. Rare and meager sets We begin with a definition and an elementary, but frequently used, observation. 1.1 Definition. A subset A of a topological space X is dense in a subset B of X if and only if its closure A con tains B. A set dense in X is said to be d ense . 1.2 Proposition. If a subset A of a topological space is dense in an open set U, then their intersection U 0 A is 8

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9 dense in U; moreover (U n A) = U . Proof. Clearly (U fl A)“ is contained in U , and so we need only show that (U n A)”” contains U. Suppose x is in U, and N is any neighborhood of x. Since x e U C A , and N n U is a neighborhood of x, N fl U meets A. Thus N meets U n A, and so x is in (U fl A) .// 1.3 Definition. A subset A of a topological space X is r are in X, or, simply, rare , if and only if its closure A contains no non-empty open set of X. A term often used for rare is nowhere dense . An example of a rare subset of a topological space is a proper closed linear subspace F of a linear topological space E. For if F contains a non-empty open set U, then, with x in U, U x is contained in F and is a neighborhood in E. Thus E = U n(U x) C F C E, and so E = F . n If a set A is not rare, then its closure contains a nonempty open set U, and so A is dense in the non-empty open set U. The converse holds as well. The following records this characterization of non-rare sets. 1.4 Proposition. A subset of a topological space is not rare if and only if it is dense in some non-empty open set.

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10 1.5 Definition. A subset A of a topological space X is rare in a subset B of X if and only if their intersection A n B is rare in the subspace B. 1.6 Definition. A subset A of a topological space X is meager in a subset B of X if and only if A is contained in a countable union °f subsets A^ of X, each rare in B. A set meager in X is said to be meager . We observe that a set A is meager in a set B if and only if A n B is meager in the subspace B. A meager set is also said to be of the first category , while a non-meager set is of the second category . Several elementary properties of meager sets follow from corresponding properties of rare sets. For this reason they are given together in the following three propositions. 1.7 Proposition. If a rare [meager] subset of a topological space contains a set A, then A is rare [meager]. Proof. Suppose a set B contains A. If B has empty interior j then the subset A of B has empty interior, and so A is rare. If B is contained in U^A^ with each A^ rare, then clearly, A is contained in U^A^ as well, and so A is meager .//

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11 1.8 Proposition. If a set A is rare [meager] in a set B, then their intersection A fl B is rare [meager]. Proof. We may assume A is contained in B. Suppose A is dense in an open set U. If A is rare in B, then, since U n B is contained in A PI B, we have U fl B empty, and so u n A is empty. From proposition 2, U = (U 0 A) = 0 } so U is empty. Hence A is rare. If A is meager in B, then A is contained in some countable union U A with each A rare in B. Thus each A D B n n n is rare. Now A n B is contained in (A^ fl B) and so is meager . // A rare set need not be rare in itself. For example, a singleton of the real line is rare, but is not even meager in itself. The next three propositions concern inheritance of the rare and meager properties. 1.9 proposition. A subset A of a topological space is rare [meager] in an open set U if and only if A n U is rare [meager] . Proof. Proposition 8 gives one direction for both properties. Conversely, let A fl U be rare, and suppose V is an open set. Now if the open set V fl U of subspace U is contained in A~ n U, then A is dense in V fl U. Since V D U is open in the space, proposition 2 gives V nuC(vnUnA) C (U n A)~. Yet U n A is rare, so V fl U is empty. Thus A is rare in U.

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12 If A 0 U is contained in with each A^ rare, then by proposition 7, each A^ fl U is rare. By the above, each A^ is rare in U. Also, A\U is rare in U. Since A is contained in (A\U) IJ ^ is meager in U.// From this, we easily see that a meager s.et is meager in 0 Vsry open set. With additional restrictions a meager set is meager in itself. As the following shows, these two properties are equivalent for dense sets. lolO Proposition. If a set A is dense in a topological space, then A is meager in itself if and only if A is meager. Proof. If A is meager in A, then, by proposition 8, A is meager. Conversely, suppose A is contained in with each A^ rare, and let U be an open set. If U fl A is contained in A^~ n A, then, since A is dense in U, we have U C (U n A)~ C (A^“ n A)” C A^“. Since A^ is rare, U is empty. Thus U fl A is empty, and so A^ is rare in A. Hence A is meager in itself.// We may localize this property as shown in the following. 1.11 Corollary. If a set A is dense in an open set U, then A is meager in A fl U if and only if A is meager in U. Proof. By proposition 2, A n U is a dense subset of subspace U. By the above proposition, A fl U is meager in itself if and only if A n U is meager is subspace U.

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13 Thus A is meager in A n U if and only if A is meager in U.// 1.12 Proposition. The collection G of rare [meager] subsets of a topological space is closed under intersections and finite [countable] unions. Proof. Since a subset of a rare [meager] set is rare [meager], G is closed under intersections. If A and A* are rare, and A U A' is dense in an open set U, then U rs contained in (A U A')“ = a“ U A'~Now the open set U\A' is contained in the rare set A , and so U\A' is empty. Thus U is contained in the rare set A' . Hence U is empty, and A U A' is rare. By induction, a finite union of rare sets is rare. Suppose A^ C U^A^^ with each A^^ rare. Now U^^A^ is contained in the countable union and so the countable union U^A^ of meager sets is meager.// This proposition provides a convenient characterization of meager sets. 1.13 Proposition. A subset of a topological space is meager if and only if it is the union of an increasing sequence of rare sets. Proof. If A is contained in U^^A^ with each A^ rare, then A equals (A U . . . U A^) H A . By proposition 12, A^ U . • . U A^ is rare, and so (A^_U ... U A^) n A is rare. The converse is clear.// • • •

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14 Section 2. Baire spaces The next definition follows Bourbaki [2] . 1.14 Definition. A topological space X is a Baire space if and only if each non-empty open subset of X is non-meager. There are several convenient characterizations of the Baire property which use the observation that the complement of an open dense set is a closed rare set and vice-versa. 1.15 Proposition. For a topological space X^ each of the following is equivalent to the Baire property: (i) Each countable intersection of dense open sets rs dense. (ii) The complement of each meager set is dense. Proof. If ^ sequence of dense open subsets of the Baire space X, then each = X\U^ is closed and rare. If U is a non-empty open set, then U is not contained in the meager set U^A^. Thus U fl = U fl ~ U\UnAn f dense. For (i) implies (ii) , suppose A is contained in U^^A^ with each A rare in the topological space X. Each = X\A^ is a dense open set. Now X\A 13 X\U^A^ n U . By (i) „ n U is dense, thus X\A is dense, n n n n

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15 Finally^ suppose (ii) holds^ and U is a meager open set of a topological space X. Now X\U is dense, yet it is disjoint from the open set U. Hence U is empty, and X is a Baire space.// The Baire property satisfies several permanence properties which we investigate in this and following sections. Proofs of the following proposition may be based on proposition 15. The one appearing here was chosen since it uses many of the foregoing propositions. 1.16 Proposition. If a topological space X contains a dense subspace Y which is a Baire space, then X is a Baire space. Proof. Suppose U is a meager open set of X. Now YOU is meager, and so, since U is open, Y H U is meager in the subspace U by proposition 9. Since Y is dense in the open set U, Y n U is dense in U by proposition 2. Hence Y D U is meager in itself by proposition 10. Now Y 0 U is meager in Y by proposition 9. But Y is a Baire space, and Y 0 U is open in Y, thus Y fl U is empty. Since Y is dense and U is open in X, U is empty. Thus X is a Baire space.// Certain subspaces of a Buire space are themselves Baire spaces. The following gives an example. 1.17 Proposition. The Baire property is open hereditary.

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16 Proof. Let U be an open subset of a Baire space X. If V is open in and V 0 U is meager in then the open set V n U is meager in X. Since X is a Baire space, V fl U is empty. Thus U is a Baire space.// Section 3. Almost open and open mappings As shown in Frolik [2], several types of mappings preserve the Baire property. However, we shall only be conCQj^ned with ones which arise naturally in linear topological spaces . 1.18 Definition. A mapping f from a topological space X into a topological space Y is almost open if and only if, for each point x of X and each neighborhood N of x, f[N] n f [X] is a neighborhood of f (x) in the subspace f[X] of Y. 1.19 Proposition. If f is a mapping from a topological space X onto a topological space Y, then f is almost open if and only if f [U] is contained in f[U] ^ for each open set U of X. Proof. Let U be an open set of X. For each x in U, f[U]~ is a neighborhood of f(x), and so f (x) is in f [U] Thus f [U] is contained if f[U] Conversely, let x be in X, and N be a neighborhood of x. Now x is in the open set N , so f(x) is in f[N^]. Yet f[N^] C f[N^r^ C f[Nr"-. Thus f[N]~” is a neighborhood of f(x).//

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17 1.20 Definition. A mapping from a topological space X into a topological space Y is an open mapping if and only if for each open set U of f [U] is an open set of the subspace f[X] of Y. From the characterization of an almost open mapping, we see that an open mapping is almost open. An example of an open mapping is the canonical mapping of a linear topological space onto its quotient by a linear subspace. . The restriction of a continuous open mapping may not be an open mapping even if the mapping is, in addition, a linear mapping which is restricted to a dense linear subspace of a normed space. This is the crux of the following. 1.21 Example. Let M be a proper dense linear subspace of an infinite dimensional Banach space B with N a closed linear subspace which is an algebraic supplement to M, that is, B = M © N. Such subspaces exist. For example, let M be the null space of a linear function g, discontinuous on B, and N be sp({x}) where g (x) 0. Let f be the canonical mapping of B onto the quotient space B/N. Since N is a closed linear subspace of a Banach space, B/N is a Banach space and so is complete. Now f is a continuous open linear mapping, yet the restriction of f to M is not open. Otherwise, since g = f]j^ is injective and continuous, g would be a topological isomorphism of the incomplete space M with the complete space B/N.//

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18 Almost open mappings are better behaved in this respect. 1.22 Proposition. If f is a continuous almost open mapping from a topological space X into a topological space Z, and Y is a dense subspace of X, then f restricted to Y is an almost open mapping from Y into Z. Proof. Let X be in Y, and U be an open neighborhood of X in X. We need to show f [U 0 Y]“ n f [Y] is a neighborhood of f(x) in f[Y]. By continuity of f and proposition 2 , f[U n Y]“ 3 f[(U n Y)“] D f[U], so that f [U 0 Y] contains f[U]"~. Now, since f is almost open, f[U] 0 f[X] is a neighborhood of f (x) in f[X], and so f[U] 0 f [Y] is a neighborhood of X in f[Y]. Yet f [U n Y]" n f [Y] contains f [U]~ 0 f[Y], and so is a neighborhood of x in f[Y].// Section 4. Some permanence properties of Baire spaces In this and following sections, we consider permanence properties of the Baire proper tyo 1.23 Proposition. The Baire property is preserved by continuous almost open mappings. Proof. Let f be a continuous almost open mapping from a Baire space X onto a topological space Y. If A is rare in Y, then f“^[A] is rare in X. For suppose f"^ [A] is dense

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19 in an open set U of X. By continuity, f[U] C f[f"^[A]~] C f[f"^[A]]" = a“, and so f [U]~ is contained in A . Since f is almost open, proposition 19 gives f[U] C f[U] . Thus, if U is nonempty, f[U] is not rare. Yet f[U] is a subset of the rare set A. Hence U is empty. Now if V is an open set of Y contained in U^A^ with each A rare in Y, then f ^ [V ] is contained in U f [A ] n f ^ [A ] rare in X. Thus f [V] is meager in the Baire space X. Since f is continuous, f ^[V] is open in X, so f“^[V], and therefore V, is empty. Hence Y is a Baire space.// 1.24 Corollary. If f is s continuous almost open mapping from a topological space X into a topological space Z, and Y is a dense Baire subspace of X, then f[Y] is a Baire space. Proof. From proposition 22, the restriction g of f to Y is almost open. Since g is continuous, the above proposition shows that f[Y] = g[Y] is Baire space.// 1.25 Corollary. If a linear topological space E is a Baire space and F is a linear subspace of E, then the quotient space E/F is a Baire space. Proof. The canonical mapping f from E onto E/F is a continuous open mapping, and the proposition shows that E/F = f[E] is a Baire space.//

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20 Under certain conditions on a non-empty topological space, for example homogeniety, the Baire property is equivalent to non-meagerness of the space. In particular and for simpler reasons, this equivalence holds for linear topological spaces. 1.26 Proposition. A linear topological space is a Baire space if and only if it is not meager in itself. Proof. Let E be a linear topological space. If E is a Baire space, then the non-empty open set E is not meager. Conversely, suppose U is a non-empty open set. If x is in U, then U X is a neighborhood in E, and so is absorbing. Thus E = U n{U x) . If E is non-meager, then by proposition 12, some n (U x) is non-meager. But f (y) = (l/n)y + x for each y in E is a homeomorphism of E, so U = f [n (U x) ] is non-meager, and E is a Baire space.// The next corollary characterizes linear topological spaces which are Baire spaces. It serves as a model for defining the new types of spaces studied in the following chapters. The proof follows immediately from the characterization of meager sets in proposition 13. 1.27 Corollary. A linear topological space is a Baire space if and only if it is not the union of an increasing sequence of rare sets.

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21 1.28 corollary. A dense linear subspace F of a linear topological space E is a Baire space if and only if F is not meager in E. Proof. From the proposition, F is a Baixe space if and only if F is not meager in F. Since F is dense, proposition 10 shows that F is non-meager in itself if and only if F is not meager in E.// Section 5. A Baire category theorem Concerning existence of non-meager sets, Kelley and Namioka [1, p. 85] state: "In spite of a great deal of work on the subject, there is essentially only one method known for showing that a set is of the second category." Cullen [1] , for instance, records four Baire category theorems, each essentially proved by the construction of a convergent sequence. However, only relatively recently have inclusive techniques been published. Such techniques are found in DeGroot[l], Frolik [1], and Oxtoby [1]. The last technique is reflected in the proof of the following classic Baire category theorem. 1.29 Theorem. A complete pseudo-metric space is a Baire space. Proof. Suppose (U^) is a sequence of open dense subsets of the space. Let U be a non-empty open set. We shall show that n U meets U, and so 0„U is dense. Thus, from proposin n n n tion 15, the space is a Baire space.

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22 Denote by 51^ the collection of all non-empty open balls V^{x) = [y e X:d(x,y) r, for some positive r and some m, implies d(x^,x) > r for all n s ra, a contradiction. Hence dist(B^,x) = 0 for all m, and so X is in each b“ . Therefore x is in n^^B^ = and n B is non-empty, n n For notational convenience, let Bq = U. Since is a dense open set, Bq n is a non-empty open set. Choose B in B, with bT contained in Bq 0 U^. Suppose B^ 6 51^, 1 ^ k ^ n, are chosen with B^ C B^_^ n U^. Since is a dense open set, B^ n ^ non-empty open set, and there ^ Vl Vl "n ^ ^n-.r of terms B^ e with B^ D B^ fl ^ so that contains the non-empty set n^B^ ^ Bq = U. Thus fl^U^ meets U.//

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23 Section 6. Pseudo-completeness of Oxtoby De Groot [1] has shown that the property of subcompactness defined by him together with regularity implies the Baire property. His proposition has, as corollaries, the standard Baire category theorems. Furthermore, he has shown that subcompactness for a metrizable space is equivalent to existence of a metric which induces the topology and for which the space is complete. We show that the pseudo-completeness property of Oxtoby [1] does not have this equivalence. We note here that the property of pseudocompleteness is satisfied by the hypotheses of the standard Baire category theorems and is implied by subcompactness with regularity. 1.30 Definition. A topological space X is quas i — regular if and only if each non-empty open set contains the closure of a non-empty open set. 1.31 Definition. A collection 51 of non-empty open sets of a topological space X is a pseudo-base for X if and only if each non-empty open subset contains an element of 55* 1.32 Definition. A topological space X is pseudo-complet e if and only if it is quasi-regular and there is a sequence (fi ) of pseudo-bases for X such that is non-empty if B e n 3 and B D) B , n n n+1 , n e w.

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24 It is clear that, in contrast to completeness for a pseudo-metric space, pseudo-completeness is a topological invariant. 1.33 Proposition. A complete pseudo— metric space is pseudocomplete . Proof. As in the proof of theorem 29, denote by the collection of non-empty open balls (x) = {y e X:d(x,y) < r} , for X e X, 0 < r < 1/n, of a complete pseudo-metric space (X,d). For a non-empty open set V, we found an element B of B with B contained in V. Thus X is quasi-regular, and n each is a pseudo-base. Moreover, for B^ e with B D b“ .we showed that n B is non-empty. Thus X is n n+1" n n pseudo-complete.// A metrizable space X is said to be topologicall y complete if there is a metric d inducing the topology of X for which (X,d) is a complete metric space. A set is said to be a Q -set if it is the intersection of a countable family of — 6 open sets. A metrizable space X is said to be an absolute G -set if X is a G -set in every metrizable space in which —6 6 it is embedded as a subspace. It is an exercise in Kelley [1] , that a metrizable space is topologically complete if and only if it is an absolute G^-set. The following example shows that a pseudo-complete metric space need not be topologically complete.

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25 1.34 Example o Let X C R be the union of the open upper half plane S = R x (0^°=) and the rational points Q x {0} on the boundary of S. Let X have the subspace topology induced by the Euclidean metric d on R . We shall show that 2 X is pseudo-complete, but not a G -set of R , and so, not o topologically complete. Let (x) = {y e X:d{x,y) < r}, for x e X, r > 0, and let = [V^(x):x 6 Q X (0,oo) , 0 < r < 1/n, S contains V (x) } . We note that if V (x) is in iB , then its closure r r n^ (x) is contained in X. Thus for with H X, we have B 3 B~ , , and so n B is non-empty as shown in the n n+1 n n proof of theorem 29. We need to show X is quasi-regular and each is a 2 pseudo-base for X. If an open set U of R meets X, then U . . 2 meets the open set S, since X C S . But Q x R is dense in R , so Q X R meets U n S. Suppose x is in (Q x R) H U fl SCQ x (0,«>) , then there is 0 < r < 1/n with and U n X D U n s D (x) D {y e R :d(x,y) 5 r/2} D " so U n X contains the closed set ^ ^r/2^^^' X is quasi-regular, and, since ''^n ^ pseudobase for X. Hence X is pseudo-complete. 2 If X is topologically complete, then X is a G^-set of R . Hence Q x {0} = X 0 (R X {O}) is a G -set of the subspace 0 R X {0}, and so Q is a G^-set of R. If Q = where each U is open in R, then U is dense in R, and R\U is rare, and so meager. Q is a countable union of singleton sets, and so is meager. Thus Uj^ (R\U^) U Q is meager. Yet

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26 U^(R\U^) U Q = (R\n^U) u Q = Rj and so R is meager, a contradiction.// 1.35 Theorem. A pseudo-complete space is a Baire space. Proof. Let V be a non-empty open subset of a pseudocomplete space X with (J5^) as in definition 32. From quasiregularity, there is a non-empty open set W with closure W contained in V. Since is a pseudo-base, there is a B in B with B contained in W. Thus B C W C V. The remainder n of the proof consists of the first and last paragraphs of the proof of theorem 29.// Section 7. Productivity of the pseudo-complete property Assuming the continuum hypothesis, Oxtoby [1] gives an example of a completely regular Baire space whose product with itself is meager. An open question cited there is the following: Is the product of two metrizable Baire spaces a Baire space? We note the question: Is the product of two or more linear Baire spaces a Baire space? In contrast to the Baire property, pseudo-completeness is productive. With slight changes we record the proof of Oxtoby [1] . 1.36 Theorem. A product of pseudo-complete spaces is pseudocomplete .

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27 Proof. Suppose are pseudo-complete spaces and for each i in I , ^^ni^n ^ sequence of pseudo-bases for X satisfying definition 32. We may suppose is in each Define R = (ll-rB-tB. e }3 B. = X. for all but a finite n '•111 ni-’ 1 1 number of i in l], so that is a collection of non-empty open sets of the product space X = H^X^ . If V is a non-empty open subset of X, then there is an open set contained in V with each a non-empty open subset of X. , all but a finite number of which equal X. . If U. = X. , let B. = U. , otherwise, choose B. in B . with closure 1 1 ^ 1 1 ^ ’ 1 ni bT contained in U. . Thus H^B. is in B , and n_B. C (n,.B.) = 1 1 Ii n Iili IIjBT C rijU^ C V. Hence X is quasi-regular, and each B^ is a pseudo-base for X. Suppose e with = B^^ 3 b“_^j^ = = n B~ , . . For each i, B . e B . and B . D B~ , . , so that I n+1,1 ^ ni ni ni n+l,i Vnl ^ = "I'^nSnl’ ^ == is pseudo-complete.// From this and theorem 35, it is clear that a product of pseudo-complete spaces is a Baire space. In particular, we have the following well-known result. 1.37 Corollary. A product of complete pseudo-metric spaces is a Baire space. Proof. By proposition 33, a complete pseudo-metric space is pseudo-complete. Thus, from the theorem, a product of complete pseudo-metric spaces is pseudo-complete, and so a Baire space by theorem 35.//

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28 Section 8. Pseudo-completeness in linear topological spaces In the next theorem, which gives a restriction on the utility of pseudo-completeness for linear topological spaces, we shall use a corollary of a well-known difference theorem discussed in chapter 3 . It states that the difference set A A of a non-meager Borel subset A of a linear topological space is a neighborhood. 1.38 Theorem. A linear metrizable space is pseudo-complete if and only if it is complete. Proof. Let (E,d‘) be a linear metric space with translation invariant metric d' . If E is complete, then E is complete in the metric d' . Proposition 33 shows that E is pseudo-complete. Conversely, suppose ^ sequence of pseudo-bases which satisfies the definition of pseudo-completeness. Let F be the completion of E, and d be the extension of d' to F. Now E is dense in F, and (F,d) is a complete metric space. We shall show that E equals F. Define C = {C: C is open in F, C D E e R and diam(C) < 1/m}. Let and U be a non-empty open subset of F. For x in U, there is 0 < r < l/(2m) with V (x) contained in U. Since R is a pseudo-base for E, and E is dense in F, there is an element B of contained in the non-empty open set of E, (x) fl E. Let C be an open set of F with B = C n E C (x) . Since E is dense in the open set c, C C (C 0 E)~ C (x)" C U. Also C C V^(x)"c

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29 [y e F:d(x^y) ^ r} ^ and so diara(C) ^ 2r < 2(l/(2m)) 1/m. Hence C is in so C is contained in Now^/BCCCun so is a dense open set of F. A = NOW F\A = = u^„(F\u_^^) with each F\U rare in F, so that F\A is meager in F. But ^ mn F = AU (F\A) is a Baire space, so, by proposition 12, A is non-meager in F. NOW A is a G -set and so a Borel set. From the remark 6 preceding the theorem, the difference set A A is a neighborhood in F. We need only show that A is contained in E, for then E C F = U^n (A A) C E, and so E F. Let X be in A. For each m and n, choose in containing x. For each m and n, there is m' > m with C . For assume not, then let r dist (x,F\C^^) . ... inn mn m*n+l since is open and contains x, r is positive. For all m mn • > m, X is in n C^.n+i assumption, is non-empty, and thus we have 0 < r ^ ^'^m n+1^ * V™' Hence 0 < r < !> , for all m' > m, a contradiction. We may thus select a strictly increasing sequence (m^) «ith =in n ^ “’n' n n n Now E is dense in the open set *^ni-j^k^ ®k ~ ^ E) C~ T by proposition 2. Thus ~ '^m n ^ ^ ^ *“m ,n+l ^ ^ “ lOj^k ^ ^ n he. By the pseudo-completeness of E, we know n^^B^ is a non-empty subset of E contained in ^

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30 n C , the sequence (ra ) is increasing, and diara(C ) < n m n^ n n “ 1/m . Thus n ^ = {x}, and so {x} = C E. Hence n m^n A is contained in E.// Suppose a linear metrizable space E is topologically complete. As a subspace of its completion F, E is a G^-set. But E is a Baire space and dense in F, so by corollary 28, E is non-meager in F. Hence E = E E is a neighborhood in F, and so E = F . Thus E is complete. By proposition 33, we may obtain this well— known result as a corollary of the above theorem, and the proof of the following is clear. 1.39 Corollary. A linear metrizable space E is topologically complete if and only if E is complete if and only if E is pseudo-complete . In example 34, we described a pseudo— complete metric space which was not topologically complete. The following, essentially an exercise of Kelley and Namioka [1], gives, existence of incomplete linear topological spaces v/hich are, nonetheless Baire spaces. In particular, there are incomplete normed spaces which are Baire spaces. By the corollary and theorem, such normed spaces are neither pseudocomplete nor topologically complete. 1.40 Example. Let E be a linear topological space which is a Baire space and has infinite dimension. An infinite dimen-

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31 sional Banach space suffices. Suppose H is a Hamel basis for E. Choose and index a countably infinite subset of H, and let = sp(H\[x^]). Since each element of E is a finite linear combination of elements of and ^ countably infinite setj we have E = From proposition 12, and since E is a Baire space, some E^ is not meager in E. Thus E^ is not rare, and as E^ is a linear subspace of E, it is dense in E. By corollary 28, E^ dense and not meager implies E^ is a Baire space. But E^ is a proper dense subspace of E, and so E is incomplete.// It is well known that a complete convex space need not be a Baire space, example 2.8 provides just such a space. Thus, by theorem 35, there are complete convex spaces which are not pseudo-complete. The product of an uncountable family of non— trivial Frechet spaces is not pseudo-metrizable , yet, by proposition 33 and theorem 36, such a product is pseudo-complete. Thus, there are pseudo-complete convex spaces which are not pseudometrizable. A more interesting example, which, by theorem 38, implies this, would be an incomplete linear space which is pseudo-complete. It is easy to see the closure of a pseudo— complete subspace is pseudo-complete, and, with theorem 38, a one-codimen sional linear subspace of a pseudo-complete linear space need not be pseudo-complete.

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32 Section 9. A productive class of convex Baire spaces There is yet another result of Oxtoby [i] which may be usefully specialized to convex spaces. We record it here . 1.41 Theorem [Oxtoby]. The product of any family of Baire spaces, each of which has a countable pseudo-base, is a Baire space. A convex space with a countable pseudo— base has a particularly simple characterization. 1.42 Proposition. A convex space has a countable pseudobase if and only if it is a separable pseudo-metrizable space . Proof. One way is clear. For the other, suppose H3 is a countable pseudo— base for a convex space E. By choosing one point in each element of J3, we obtain a countable dense set, and so E is separable. To complete the proof, we obtain a countable neighborhood base at 0 from j|. For a set B, define be (B) to be the smallest balanced convex set containing B. Let = (bc(B): B e B] . Suppose B is in B, and x is in B. Now the open set V = (1/2) (-x) + (1/2)B contains 0 and is contained in be (B) . Hence the countable family is a family of neighborhoods of 0

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33 For a balanced convex neighborhood U, there is an element B of B contained in U. Hence be (B) is contained in U, and !Jl is a neighborhood base at 0.// Clearly, theorem 41 reduces to the following for convex spaces . 1.43 Theorem. A product of any family of convex Baire spaces, each a separable pseudo-metrizable space, is a Baire space. Sectio n 10. Inheritance of the Baire property A long-standing problem noted by N. J. Kalton, who attributes it to V. K. Klee and A. Wilansky, is to prove or disprove the following; 1.44 Conjecture. A non-zero linear functional on a Banach space is continuous if and only if its null space is meager.* Of course, it is well known that a linear functional on a linear topological space is continuous if and only if its null space is closed. This may be recast in slightly different terms. Since the null space of a non-zero linear functional is a linear subspace of codimension 1, it is closed if and only if not dense, and not dense if and only if rare. Thus a non-zero linear functional is continuous if and only if its null space is rare. This is related to the conjecture above, and both are connected to the following. * N. J. Kalton, pers onal communication to S. A. Saxon

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34 1.45 Conjecture. A non-zero linear functional on a linear Baire space is continuous if and only if its null space is meager . As will be shown, this conjecture is equivalent to inheritance of the Baire property by linear subspaces of countable codimension. We begin with the following observation. 1.46 Proposition. Conjecture 45 is equivalent to inheritance of the Baire property by dense linear subspaces of codimension 1. Proof. Let E be a linear Baire space. Suppose the coniecture holds, and N is a dense linear subspace of codiraen— sion 1 in E. By corollary 28, we need only show that N is non— meager in E in order to show that N is a Baire space. Let X be in E\N, so E = N © sp({x}) . Define a function f from N U [x] to the scalar field by f[N] = {0} , and f(x) = 1, and extend f to E by linearity. Now f is a linear functional on E with null space N. Since N is a dense proper subspace of E, f is not continuous. By the conjecture, the null space N is non-meager. Conversely, the null space of a non— zero continuous linear function on E is a proper closed linear subspace of E. Such a linear subspace is rare, and therefore meager. Assuming

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35 the inheritance property, we need only show that a discon tinuous linear functional f on E has a non— meager null space. Let N = N(f) be the null space of f. Since f is discontinuous, N is dense. But N is also of codimension 1, and so is a Baire space. Corollary 28 shows that N is nonmeager .// The following gives a technical observation. 1.47 Lemma. Let the linear topological space E contain linear subspaces F and G with intersection {0]. Suppose F and G contain a closed set A and a compact set K, respectively. If A is rare in F, then A + K is rare in E. Proof. Suppose A + G is a neighborhood of some point in E. That is, for some a in A and g in G, A + G is a neighborhood of a + g. Clearly, (A + G g) n F is a neighborhood of a in F. For x in (A + G g) OF there are a' in A and g' in G with x=a' +g' -geF. Hence x a g' g is in F n G = [0} , so that x = a' e A. Thus 4 G _ g) fi F is contained in A, and so is a neighborhood of a in F. Now A is closed in E and K is compact, so A + K is closed in E. If A + K is not rare in E, then A + K and so A + G is a neighborhood of some point of E. From the foregoing, A is a neighborhood of some point in F . So A is not rare in F, and the contrapositive gives the lemma.//

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36 The following example shows that the requirement in the lemma that A be closed in E is important. 1.48 Example. Let E be an infinite dimensional normed linear space with closed linear subspace L of codimension 1. Suppose H is a Hamel basis for L with each element of norm 1, X is in E\L, and (x^) a sequence of distinct elements of H. Define a scalar-valued function f on H U {x} by n; u X n f (u) otherwise. and extend f to E = sp(H U {x}) by linearity. Now f is a linear functional on E. Since (x^) is contained in the linear subspace L, we observe that x^/n converges to 0 in L, since llxynjl = Hx^H/n = 1/n converges to 0, Yet f(xyn) = f(x ) /n = n/n = 1 does not converge to 0. Thus f is discontinuous on L, and so on E. From this, the null spaces N(f) and N(fjj^) = L n N(f) are proper dense linear subspaces of E and L respectively. Let F = N(f). It is easy to show that E\{F U L) is non-empty, so let y be in E\(F U L) . Now the intersection of linear subspaces F and G = sp([y}) is {O}, in fact, E = F © G. Let A = L n F c F, and K = [ay: \a\ S 1} C G, so that A is closed in F and K is compact. Since L is a closed linear subspace of E, A = L D F is a closed linear subspace of F. If A = L fl F = F, then

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37 F C L. But F = N(f) is dense and L is closed in E, so L = E. This is a contradiction since L is of codimension 1. Thus A = L n F is a proper closed linear subspace of F, and so A is rare in F. However A + K is not rare in E as is now shown . Since A~ is closed in E and K is compact, A + K is closed in E. From this and continuity of addition, a“ + K D (A + K)~ D A~ + K = A + K, so (A + K)“ = a“ + K = (L n F)~ + K = L + K, which is a weak neighborhood of 0. For define g [L] = {O}, and g(y) = 1 and extend g to E = L © G by linearity. Since the null space N(g) = L of g is closed, g is continuous, moreover, (A + K)~ = L + K = [u e E: ]g(u)] ^1}. Hence A + K is not rare in E.// We recall that an F^-set of a topological space is the union of a countable family of closed sets. 1.49 Theorem. The Baire property is inherited by F^-linear subspaces of countable codimension. Proof. Let E be a linear Baire space. Suppose F is a proper linear subspace of countable codimension in E, and F is an F -set of E. We shall show that F is nona meager in itself. Let (A^) be an increasing sequence of closed subsets of F with F =

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38 Since F is an F -set of there is an increasing a sequence (C ) of closed subsets of E with F = Now F = ^ ^n^ • since C F is closed in E and A C F is closed in Fj n (^'n " = *n ^ ^ “ <*n ^ F) n = = A n c , n n^ so A n C is closed in E. Finally^ if A (1 C is not rare n n u 11 3 _n F, then A^ is not rare in Fo For notational convenience, we may assume each A^ is closed in E. Since F is a proper linear subspace of E and of countable codimension, there is a sequence (x^) in E such that E = F © sp({x^}^) . Letting G = sp({x^]^) , we have F H G= {0} . Each A is contained in F and closed in E; each K = n “ joj^I ^ n for 1 5 k n} is contained in G and compact. Now E = ^ E is a Baire space. Thus some A + K is not rare in E, and so, by the lemma, n n A is not rare in F.// n It is not difficult to see, independent of the above, that a closed linear subspace of countable codimension in a linear Baire space is of finite codimension. This is included in the following. 1.50 Corollary. An F —linear subspace of countable codimen sion in a linear Baire space is closed and of finite codimension .

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39 Proof. Suppose F is a linear subspace of a linear Baire space E, and is a sequence in E with E = F + sp({x^}^). If F is closed, then each E^ = F + sp({xj^]^) is closed. Yet E = H^E^. As E is a Baire space, soine E^ is not rare. Since a proper closed linear subspace is rare, E = E^ = F + sp({x^}j^), and so F is of finite codimension in E. We need only show that F is closed if F is an F subset of E. a Let (C ) be a sequence of closed subsets of E with F = II C . From the theorem, F is a Baire space, so that ^ n n some is not rare in F. Thus for some open set U, C OF contains U fl F 7 ^ 0. But F is dense in the open n _ set U 0 f"~ of the space F~, so by proposition 2, (U n F) contains U fl F . Now F D D (U fl F) D U fl F . Since U fl F~ is a non-empty open subset of F , F is F , and so closed.// Finally, we have the following. 1.51 Proposition. Conjecture 45 is equivalent to inheritance of the Baire property by linear subspaces of countable codimension. Proof. Let F be a linear subspace of countable codimension in a linear Baire space E. Let (x^) be a sequence in E with E = F + sp({x^}^). Letting E^ = F + sp([x^}^), we have E = ^n^nÂ’ E is a Baire space, some E^ is not meager in E. By corollary 28, E^ is a Baire space.

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40 Now F is of finite codimension in = F + sp({x^}^). Also F~ n E is a closed linear subspace of Baire space n E and of finite codimension in E . Hence F n E is a n n 11 Baire space by theorem 49. Since F is of finite codimension in F~ n and dense in F~ n E^, we may assume that F is of finite codimension in E and dense in E. From finite induction, inheritance of the Baire property by dense finite codimensional linear subspaces is equivalent to inheritance by dense 1-codimensional linear subspaces. Proposition 46 completes the proof.//

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CHAPTER 2 Convex Baire Space Analogs In this chapter we generally restrict our attention to convex spaces, and consider for these spaces several properties related to the Baire property. Relations among these properties are discussed, and permanence of these properties under certain operations is found to be stronger than presently known for the Baire property itself. In particular, the properties discussed are productive and inherited by linear subspaces of countable codimension, both permanence properties which are, as yet, unknown for the Baire property. Section 1. The new convex spaces From corollary 1.27, we obtain the following observation: A convex space is a Baire space if and only if it is not the union of an increasing sequence of rare sets. Using this characterization as a model, we may obtain more general properties which we now define. 2.1 Definition. A convex space is Baire-like if and only if it is not the union of an increasing sequence of balanced convex rare sets. 41

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42 2.2 Definition. A barrelled space is quasi-Baire if and only if it is not the union of an increasing sequence of rare linear subspaces. By relaxing the inclusion requirement, we obtain more restrictive properties. 2.3 Definition. A convex space is unordered Baire-like if and only if it is not the union of a countable family of balanced convex rare sets. 2.4 Definition. A barrelled space is unordered quasi-Baire if and only if it is not the union of a countable family of rare linear subspaces. The barrelled property may be cast in similar terms as follows: A convex space E is barrelled if and only if it is not the union of a sequence (nB) where B is a balanced convex rare set. For if E is barrelled and E = U^nB, then B is absorbing, and so B~ is a barrel in E. As E is barrelled, B is a neighborhood in E, and so B is not rare. Conversely, if B is a barrel in E, then E = U^nB, so that B is not rare. B = B is balanced and convex, B is a neighborhood. Thus E is barrelled. From inspection of the definitions above, the implications in the following diagram are clear for convex spaces. Examples in the next section establish the remaining relationships.

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43 2.5 Diagram. Baire unordered Baire— like unordered quasi Baire ^ if i\ Baire-like quasi-Baire ==:^ barrelled <•= It is not yet known whether or not the Baire and unordered Baire-like properties are distinct. Section 2. Distinguishing examples Amemiya and Komura [1] have shown that a barrelled and metrizable space is Baire-like. Saxon [1] has generalized this to the following. 2.6 Theorem. If a convex space E is barrelled and does not have a linear subspace topologically isomorphic to cp, then E is Baire-like. Since cp has countably infinite dimension, cp is a union of an increasing sequence of finite dimensional subspaces each closed, since cp is Hausdorff, and so rare also. Thus cp is not a Baire space or even a quasi-Baire space. Since each balanced convex absorbing subset of cp is a neighborhood, cp is a barrelled space. Therefore cp distinguishes between barrelled and quasi-Baire spaces.

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44 Each linear subspace of a convex space with the strongest convex topology is closed. Since the completion of such a space has the strongest convex topology, such a space is complete. Thus, if cp is metrizable, it has a translation invariant metric d, and (cp,d) is a complete metric space. By the Baire category theorem, this is a contradiction. Hence is not metrizable, and the above theorem implies that barrelled metrizable spaces are Baire-like. We shall use this in the following example which is similar to an example in Saxon [1] , much simplified. The example provides a Baire-like space which is not unordered quasi-Baire. 2.7 Example. Let be the space of all real sequences with the product topology. Let e^ = e R , and, interpreting summations coordinate-wise, let E = (n^) strictly increasing, lim^ (n^/k) = <», e R} . Observe that E is a linear subspace space R^. For suppose f E with their sum, where JC o of the convex metrizable' E, 0, e are elements ^ k qk (n^) is the strictly increasing sequence formed of the elements of ^’^k^keo)* For each k, let i,j be the largest integers such that P^,
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45 E is not unordered quasi-Baire. For suppose is the projection of in the nth coordinate, then g^ is a continuous linear functional which is non-zero on E. Thus N(g^) n E is a rare linear subspace of E. Clearly, E = U^N(g^) n E. We shall now show that E is barrelled, and so, from the preceding remarks, E is Baire-like. To show that E is barrelled, we need only show that each pointwise bounded subset A of E' is equicontinuous , Since E is dense in we may consider E' to be all of (R^) ' = SjjjjR. By density of E, A° is a neighborhood in r“^ if and only if A° n E is a neighborhood in E. Thus A is equicontinuous on R*^ if and only if A is equicontinuous on E. Now A is equicontinuous if and only if A is bounded at each e and there is a natural number m such that n Afe ] = fO] for all n s: m. . Assuming A is not equicontinuous, we need only find x in E at which A is unbounded in order to show E is barrelled. If A is unbounded at some e^, we are done. Suppose A is bounded at each e^. CLAIM; There are strictly increasing sequences (m^) , (n^) of natural numbers and sequences (f^) in A and {a^) is R such that (a) ^ ° ™ ^ P 2 and (c) n^^ 3 ^ > (k + 1) , n^, m^. For, since A is not equicontinuous, there are f.^ in A and n, with f^ (e^ ) ^0. Select with f ^ ) = 1. There 1 X ]_

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46 is rn with f, (6 ) = 0 foir in s in. Since A is not equi— lira Jcontinuous , but is bounded at each e^, there are £2 iri A establishes the claim. From (c) , (n^) is strictly increasing and n^/k > k, , ’ of E. Now from the claim and continuity of f^. Thus A is unbounded at X in E.// To show the remaining implications, excluding the first, are strict, we need an unordered quasi-Baire space which is not Baire-like. The following, suggested by S. A. Saxon, gives such an example. In addition, we shall note that the space constructed is complete. 2.8 Example. Topology aside, is an increasing sequence of linear subspaces of R^, and so E = is a linear space. Supply E with the strongest convex topology for which each injection ij^: (E^7) is continuous. Let Sj^ be the closed unit ball of 1!^^) • Now nS^ is contained in (n+l)S^^^, and so E = U^nS^Clearly E is not Baire-like if each S^ is rare in E. We need only show E is Hausdorff and each S^ is complete. For, if so, S^ is closed and, since S^ is balanced and con2 and ^2 larger than 2 , n^ and m^ su Select 02 so that f 2 ^*^l®n ^2®n ^ 1 2 such that f ^ (e ) ^ 0, 2 n2 ) = 2. An induction e ) + f n P

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47 1 ; • vex S is rare if and only if S is not a neighborhood in E. •’ n “ Yet S is not absorbing j so S is not a neighborhood in E. n “ For X in E. let f, (x) be the kth terra of the sequence x. JC since f, is a linear functional and each continuous , iC f is in E' . If X is a non-zero eleraent of E, then sorae f^ Ic is not zero at x. Thus E is Hausdorff. Now suppose x^ = ^^ra^ ^ Cauchy net in E. Since f^ is continuous, we raay let = lira^f^(x^) = lira^cXj^^™^ and x = (ct^) • each p, “ ^^'^ra ^^=1 ^ *^p ^ by continuity of ] f^ ( • ) 1 ^ • Yet M ^ ^ l]x^!iJJ ^ 1, so ^ 1^ in S . Hence S^ is complete, n n We note E is barrelled and the union of an increasing sequence (nS^) of balanced convex complete sets. From Valdivia [1, Th. 1], E is complete, while, from the above, E is not Baire-like. Now E is unordered quasi-Baire. For if E = Uj^E^ where each E^ is a closed linear subspace of E, then ~ '^n ^^n '^1^ ^ and E n is a closed linear subspace of Since the norm topology on is stronger than the inherited topo2^ogy, each E^^ fl is closed in the norm topology. Since 11]^) ^ Baire space, some ^ Yet is dense in E since Ij^ contains the terminating sequences which are dense in each • Thus E = (E^ 0 l-^) ^ E^, and so e“ = E. Hence E is unordered quasi-Baire.//

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48 Finally^ we remark that the barrelled requirement in definitions 2 and 4 is not redundant. For suppose B is an infinite dimensional Banach space with the weak topology a, then the weak topology for B is strictly smaller than the norm topology, and sc (B,a) is not barrelled. Yet E is not the union of a countable family of linear subspaces rare in (E,a) • Section 3. A characterization of unordered Bair e-like spaces We shall see from proposition 3.5 that each of the four new properties is characterized by requiring a barrelled space not to be the union of translates of members of any family of sets described in its definition. A particularly useful characterization of the unordered Baire-like property has been noted by S . A. Saxon and is recorded here. 2.9 Proposition. A convex space is unordered Baire-like if and only if it is not the union of a countable family of linear subspaces each either rare or not barrelled. Proof. Suppose E = each F^ is rare in E or not barrelled. In the former case, B^ = F^ is clearly a balanced convex rare set. In the latter, suppose B^ is a barrel in F^ which is not a neighborhood in F^. In this case as well, B^ is rare in E. For otherwise, since B^ is balanced and convex, B^ is a neighborhood in E, and so

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49 B = B n F is a neighborhood in F , a contradiction, n n n n Finally E = U ® unordered Baire-like. ^ ’ mn n Conversely^ suppose E = and each B^ is balanced and convex. Let F^ = sp(B^)j so that E = U^F^. Some F^ is both barrelled and dense. Now B^ PI F^ is a barrel in F^, and so a neighborhood in F^. Hence B^ is not rare in F^. Since F^ is dense in E, B^ is not rare in E. Thus E is unordered Baire-like.// We note here that a non-barrelled subspace F of a convex space E is meager in a rather strong sense. For suppose B is a barrel in F which is not a neighborhood in F. Since B is balanced and convex, B is rare in F. Thus each nB is rare in E, yet F = U^^nB. Section 4. Some permanence properties Many of the permanence properties enjoyed by Baire spaces hold for the several related convex spaces. 2.10 Proposition. If a convex space E contains a dense linear subspace F which is (unordered) quasi-Baire [Bairelike] , then E has the same property. Proof. Suppose (B^) is a sequence of balanced convex sets, and E = U^B^Now (B^ n F) is a sequence of balanced convex subsets of F, and F = (B^ H F) . If

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50 crsasing or consists of linoair subspacos^ the ^ has the corresponding properties. Now if n F is not rare in then, since F is dense in E, PI F, and hence B , is not rare in E.// n In the context of linear mappings the concepts of almost open and open mappings have particularly simple forms. We record them in the following. 2.11 Proposition. A linear mapping f from a linear topological space E into a linear topological space F is open [almost open] if and only if for each neighborhood U in E, f[U] [f[U]"~n f[Ej] is a neighborhood in f[E]. Proof. One direction is immediate from the definitions. We prove the other for almost open linear mappings. Let X be in E, and U be a neighborhood of x. Now U x is a neighborhood in E, and so f[U x] n f[E] is a neighborhood in f[E]. The closed set f[U]“” f (x) contains f[U] f (x) = f[U x] , and so contains f[U x] . Thus {f[U] f(x)) flf[E] is a neighborhood in f[E], and hence f[U] n f[E] is a neighborhood of f(x) in f[E]. Thus f is almost open. The proof for open mappings is similar and simpler.// The following is a standard context in which almost open mappings arise. 2.12 Proposition. A linear mapping from a convex space onto a barrelled space is almost open.

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51 Proof. Let E be a convex space and U be a barrel which is a neighborhood in E. If f is a linear mapping of E onto a barrelled space F, then f[U] is balanced, convex and absorbing in F. Hence f[U] is a barrel in F, and so a neighborhood in F. By proposition 11, f is almost open.// 2.13 Proposition. The (unordered) quasi-Baire [Baire-like] property is preserved by continuous almost open linear mappings . Proof. Let f be a continuous almost open linear mapping from a convex space E, enjoying one of the above properties, onto a convex space F. F is barrelled, for if B is a barrel in F, then, by linearity, f ^[B] is balanced, convex, and absorbing in E, Since f is continuous, f~^[B] is closed, and so, a barrel in E. Thus f [B] is a neighborhood in E, and, as f is almost open, B — B — f[f“^[B]]~ is a neighborhood in F . Now if F = where each B^ is balanced and convex, then E = U f“^[B ]. By linearity, each f ^ [B ] is balanced n n and convex. If (B^) is a sequence of linear subspaces, so too is (f~^[B^]). Moreover, inclusions are preserved by f Finally, if [B^^J is not rare, then, since f [B^] is balanced and convex, f ^ [B^] is a neighborhood in E. As f is continuous, f ^ [B^^ ] is closed, and so contains f

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52 Thus f ^ [B ] is a neighborhood in E. Since f is almost open, B~ = f[f~^[B~]]~~ is a neighborhood in F, and so B^ is not rare.// The following contains an additional permanence property for convex Baire spaces. 2.14 Corollary. If an (unordered) quasi-Baire [Baire-like or convex BaireJ space E is mapped onto a barrelled space F by a continuous linear mapping f, then F has the same property as E. Proof. From proposition 12, the linear mapping f is almost open. Since f is also continuous, propositions 13 and 1.23 apply.// 2.15 Corollary. Let f be a continuous almost open linear mapping into a convex space from a convex space E containing a dense linear subspace G. If G is (unordered) quasi-Baire [Baire-like], then f[G] has the same property. Proof. By proposition 1.22, f restricted to G is almost open. Since f is also a continuous linear mapping, the proposition gives the result.// 2,16 Corollary. If a convex space E is (unordered) quasiBaire [Baire-like], and F is a linear subspace of E, then the quotient space E/F has the same property.

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53 Proof. If f is the canonical mapping from E onto E/F , then f is a continuous open linear mapping.// Section 5. Inheritance There has been some recent interest in inheritance of properties of convex spaces by linear subspaces of coimtable codimension. Evidence of this includes De Wilde and Houet [1], Levin and Saxon [1], Saxon [1,2] ^ Saxon and Levin [1], and Valdivia [1,2]. In particular, we shall use inheritance of the barrelled property shown in Saxon and Levin [1] . This inheritance property for the Baire-like and quasi-Baire properties is shown in Saxon [1,2]. For completeness, we shall record the inheritance for these two properties, and present a proof of it for the unordered properties. For the latter, we need a fact about linear spaces given by a corollary of the following. 2.17 Proposition. If the union of two countable families g: , 3 : of linear subspaces of a linear space E covers E, 3. 2 then one of them covers E. Proof. Assume x,y are elements of E/USP^^^ E\U3>2 i^sspectively. Since E = D ^ and y are distinct, and the line L passing through x and y is an uncountable subset of E. If an element F of 3^ contains two distinct points of L, then, since F is a linear space, F contains L, and so both x and y, a contradiction. Since U '^2

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54 countable^ it covers only a countable subset of the uncountable set L. Yet L C E = U (3^ U 32^ ^ contradicting the assumption.// This gives two corollaries. 2.18 Corollary. A linear space is not covered by a finite family of proper linear subspaces. Proof. Suppose no linear space is covered by fewer than n proper linear subspaces, and linear space E is covered by family 3 of n proper linear subspaces. For F in 3, F / E, and so the proposition implies that the set of n 1 proper linear subspaces 3 \{f} covers E, a contradiction.// 2.19 Corollary. If a linear space E is covered by a countable family 3 of proper subspaces, then E is covered by each cofinite subfamily of 3. Consequently, each finite dimensional subspace F of E is contained in each member of an infinite subfamily of 3. Proof. If 3-j^ is a finite subfamily of 3, then 3^ does not cover E from corollary 18. By the proposition, 3\3^ covers E. Now 5 is countably infinite, so we may index 5 uniquely as {F^: n e uj] . With its only Hausdorff linear topology, F is a Baire space, and each F^ H F is a closed subspace

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55 of F. Now if F is not contained in F^, then F^ 0 F is rare in F. Since F is a Baire space and F C U^F^, some F contains F. Suppose n, < n„ < . . . < n have been n^ 1 ^ P chosen with F C F for 1 ^ k S p. Since F C U F , n^^ n>np n there is some n ^ > n such that F contains F. This P+1 P ^p+i induction defines a sequence (F ) in I? of distinct elements each containing F.// 2.20 Theorem. The properties (unordered) quasi-Baire [Bairelike] are inherited by linear siibspaces of countable codimension. Proof. The ordered cases have been given elsewhere. Let E = F + ®P(t^n^n^ where F is a linear subspace of an unordered quasi-Baire [Baire-like] space E. Some F^ = F + sp({x^}^) is unordered quasi-Baire [Baire-like]. Otherwise each F^ = '-'m^mn each linear subspace F^^ is rare in F^ [or not barrelled] . Yet E = so some F is dense in E [and barrelled], a contradiction. Moremn over, F^ is barrelled since it is of countable codimension in the barrelled space E. F is of finite codimension in F^. Thus we need only show that the properties are inherited by linear subspaces of finite codimension. By finite induction, it is sufficient to show that the properties are inherited by linear subspaces of codimension 1. Let F be of

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56 codimension 1 in E, and E = F + sp({x}) . We note that F is barrelled. Let Q be a countable family of linear subspaces of F which covers F. Assume each element of Q is rare in F, and let . G € (j) . No element of K contains F and ii covers Fj so M is not finite by corollary 18. Elements of M are of codimension at least 1^ so for distinct elements H of H, n is of codimension at least 2. Thus 1 2 -L ^ H n + sp({x}) is a proper closed linear subspace of E, 1 2 and so rare in E. Now for y in F, the 1-dimensional subspace sp({y}) of F is contained in each of an infinite subfamily of M by corollary 19. Hence there are distinct elements H 2 of M with y contained in fl H 2 » Therefore, E = F + sp({x]) For E either unordered quasi— Baire or unordered Baire— like, we have a contradiction. In the former case, the contradiction gives the inheritance. In the latter case, we see from proposition 17, that we may suppose each element of Q is dense in F. We need only show some element of Q is barrelled. Now E = F + sp({x}) = U{G + sp({x}): G e Q} , and so some G + sp({x}) is barrelled. Hence G is barrelled.// Section 6. Productivity Productivity for the ordered properties have been discussed in Saxon [1] and [2] . Productivity for the

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57 unordered properties requires significantly different techniques from those found there. Yet these techniques also give proofs for productivity of the ordered properties. The beginning of these techniques was proposition 17. This proposition along with the following lemmas and corollaries form the basis of attack for this problem. 2.21 Lemma. If a countable family iB of balanced convex sets of a linear space E covers E, then if U B 2 = {kB: k e uu, B e B}, either or alone covers E. Proof. Let 3^ = {sp(B): Be B and covers sp (B) } , k = 1, 2. Clearly covers For B in B, (nB) is a sequence in B^_ U B 2 , so some subsequence (n^B) is in either or Now sp(B) = Uj_n^B, and so either B^ or ^2 alone covers sp(B) . Hence sp(B) is in U ^23 ^nd this covers E. Since covers either B^^ or B^ alone covers E.// 2.22 Lemma. Let (E^) be a sequence of linear topological spaces, B a countable family of closed balanced convex subsets of the product E = n^^E^. If B covers E, then the family {kB: k e ua, B e B, for some Mem, ^ b} covers E. (We identify as a subspace of the product Vm-> Proof. We may suppose that B contains all integral multiples of its elements. Let = {B e B: for some

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58 M e 0), ^ b] and = E\ 5 I^. We shall show that covers E by showing that B2 does not cover E and applying lemma 21 . If B2 is empty the result is direct. Otherwise index B2 n e uo} . CLAIM: There are a strictly increasing sequence (m^) of natural numbers, and a sequence (x^) in E such that, for each k s 1 , (b) S^x. + y / for all y e Since is a proper subset of E, let x^ be in E\B^. With m^ = 1 , (a) is satisfied for k = 1 and (b) is satisfied vacuously for k < 1 . Suppose m^^ < m2 < . . . < X X are chosen to satisfy (a) for k s n and (b) for I-* ^ n k < n. If there are y^ € so that „ n B for each i, then SV^+ converges to E,x . Since B n 131 -Lj is closed, S^x^ is in B^, a contradiction. Hence there is “n =• “n-l y ^ <*’> is satisfied for k < n + 1. NOW if c then, since is n balanced and convex, E^ C a contradiction. Thus there is x , e H E^ with S i ®n+l* claim n X _ui^ j follows by induction. Since (m^) is strictly increasing, and x^ e the coordinate-wise sums 5]^ exist and are in IIj^^j^E^.

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59 From this and (b) , for each k, x = i Hence x is not in U 3 ^B^ = UB 2 , so 8 ^ does not cover E.// We have the following as a direct application of this lemma. 2.23 Corollary. Let (E^^) be a sequence of linear topological spaces, and a countable family of closed linear subspaces of the product E = If covers E, then the collection {F e S': for some M, ^ F) covers E. 2.24 Lemma. Let (E^) be a sequence of linear spaces and 5 be a countable family of proper linear subspaces of the product E = Suppose each F in S' contains ITj^^E^ for some M e u). If 3 covers E, then for some m, the collection {F e Sf: E^ 9 Z f] covers E^. Proof. Let S'^^ = {F e 3r: E^ ^ f} . Assume for each m that r? does not cover E . We shall obtain an element x m m of E not covered by 5 . Note that S' = for otherwise assume F is in S' with E contained in F for all m. There m is M with C P. NOW E = C F, a contradiction, since F is a proper subset of E. For each m, let be in E^\IJ3^. CLAIM: There is a sequence (a^^) of scalars with For S^OiXi ^ U(S'3_U. 1 , ^ US'j^) for all m e tu. Suppose a^,...,a^ are chosen • • •

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60 so that i U for 1 ^ k ^ n, and let L be the line determined by ^n+l ' I£ two distinct points of L lie in an element F of ^ then F contains L, and so F contains both and a contradiction. Since is countable it covers only a countable subset of the uncountable set . A short computation shows there is with ^""l\^i ^ U(53_U...U3r^^3_). Now each coordinate-wise sum ^msn^rn* Let X = V a X . For F in 3, there are M and n with m m m ^>MVm ^ ^nWe may suppose M is larger than n, and so ^ U(ffj^U...U5„) so that and so X = Is not in F.// 2.25 Theorem. The properties (unordered) quasi-Baire [Baire-like] are countably productive. Proof. Let (E^^) be a sequence of convex spaces enjoying one of the properties. The product E = is barrelled. Suppose g is a countable family of closed

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61 balanced convex sets each rare in E. If each is (unordered) quasi-Baire, then the elements of B are to be, in addition, linear subspaces of E. In case R consists of an increasing sequence (B^) which covers E, lemma 22 yields natural numbers k, N and M such that kB^ contains Thus B^ contains for all n s N. Hence (B^ n is an increasing sequence M of balanced convex sets rare in G = Since R covers E, for each m there is n ^ N with B n E^ not rare in E^. ™ ^ m Let n = maxfn : 1 ^ m 5 M} . Hence B fl G is not rare in G, m n a contradiction which shows E is quasi-Baire [Baire-like] . In case the spaces {E^]^ are unordered quasi-Baire [Baire-like], the following argument is required. If R covers E, we may suppose it contains all integral multiples of its elements. Lemma 22 shows we may assume each element B of R contains II ^,E for some M. Now F = sp (B) contains m>M m n E . Moreover, since the closed balanced convex set B m>M m is rare in barrelled space E, B is not absorbing in E, and so F is a proper linear subspace of E. Thus jj = (F: f = sp(B), B e R} fulfills the hypotheses of lemma 24 This lemma gives some E^ covered by (F g 3=: E^ ^ f}. In case R consists of closed subspaces of E, 5 = R, and E is covered by a countable family of closed proper linear m subspace {F fl ^ ^ o'} ^ ^ contradiction of the un— ordered quasi-Baire property for E^.

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62 Consider F in 5 with B in B such that ^ F = sp(B). If F n E is dense in E , then the barrel B D E in E is ni in not a neighborhood in E^. Otherwise, since B 0 E^^ is closed in E , B n E is a neighborhood in E . This is a contradiction since sp(B 0 E^) = F fl E^ E^. Hence foe is not barrelled. Therefore the countable family m (F n E^: Ej^ ^ F e covers E^^, yet no F H E^ is both dense in E and barrelled, a contradiction of the unordered m Baire-like property for E^^ by proposition 9.// The following lemma is lifted out of a proof in Saxon [1] . In its proof, we shall use twice this consequence of the bipolar theorem discussed in Horvath [1]: If B is a closed balanced convex set of a convex space E, then an element x of E is in the span of B if and only if B° is bounded at x. 2.26 Lemma. If is ^ collection of barrelled spaces and B a countable family of balanced convex rare subsets of the product E = then there is a countable subset j of I with each B fl RjE^ rare in HjE^ . Proof. We may suppose I is infinite, and each element of IB is closed. Since E is barrelled, and each element B of B is a closed balanced convex set rare in E, B is not absorbing, and so E\sp(B) is not empty. Index B as ^®n' ^ ^ ^ ^n E\sp(B^).

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63 Since is not in sp (B^) , there is a sequence in B° C E' = Z El which is unbounded at x . For each nil ^ f there is a finite subset I of I with f [n,., E.] = {O}. mn rori mn x-'-^^n Let J = U I , and F = n^E.. We shall show each B D F mn ran^ J 3 “ is rare in F. Define (y^) in F by = tTjX^, where tTj is the projection of E onto F. By choice of J, = ^mn^n lo.n, and so (*„„)„ is unbounded at y^. Now ®n’ and E„ D n F, so c B° C (B„ n F)°. BUt^B„ 0 f" is a closed balanced convex set of E with (B^ OF) = (B n F)° unbounded at y . Therefore y is not in the span V n n n of 0 F~. Since y^^ is in F, B^ 0 F is not absorbing in F, and so not a neighborhood in F. Yet B^^ 0 F is closed balanced and convex in F, and so B^ n F is rare in F.// 2.27 Theorem. The properties (unordered) quasi-Baire [Baire-like] are productive. Proof. We use the notation of the previous lemma. If R covers E, then (B n F: B e a] covers F. If the cover IB contradicts one of the properties for E^ then {B 0 F: B € a] does so for the countable product F = IIjB j • Since the properties are preserved by countable products, the proof is complete.//

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CHAPTER 3 Category Analogs In this chapter, we discuss concepts in convex spaces analogous to category in linear Baire spaces. Since the analogs to the Baire property discussed in the previous chapter entail linear space concepts such as absolute convexity, the analogs to category are initially problematic. The success of the definitions adopted here is reflected in the analogs found for some standard category theorems in linear topological spaces. In particular, proposition 5 is analogous to the characterization of linear Baire spaces in proposition 1.26, Section 1. Analogs of meagerness We simplify the presentation by restricting the discussion to the context of the Baire-lihe property. Only slight changes in wording are necessary for the other properties, and, in the cases of the unordered properties, the proofs are generally shortened. 3.1 Definition. A subset A of a convex space E is meagerlike in a subset B of E if and only if there are an increasing sequence (B ) of balanced convex subsets of E and a sequence 64

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65 (x ) in E such that A is contained in (x + B ) and each ' ' n xi X + B is rare in B. A is said to be meager-like if A is n n meager-like in E. Similar definitions may easily be formed using [ unordered ] quasi-meager and unordered meager-like , and, with minor changes, for example, the insertion of the barrelled property, the following theorems apply to all of these properties. The theorems stated in this and the next section are analogs of standard category theorems for linear topological spaces which may be found in Kelley and Namioka [1] . These theorems may be obtained by replacing "Bair e-like" by "Baire," "meager-like" by "meager," and "convex space" by "linear topological space." Since there are Baire-like and unordered quasi— Baire spaces which are not Baire spaces, these theorems extend category-like structures to some spaces about which the standard category theorems provide no information. 3.2 Proposition. Translates and scalar multiples of meagerlike subsets of a convex space are meager-like. Proof. Suppose A C (x^ + B^^) and each x^ + B^ is rare in E. Thus y + a(x^ + B^) is rare in E, for y an element of E and a a scalar. But y + oAC ( (y + ax^) + aB^) .// The proof of the following is clear.

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66 3.3 Proposition. If a set A is meager-like in a subset B of a convex space ^ then each subset of A is meager-like in B. For the [unordered] quasi-Baire case we require E to be barrelled in the following propositions. 3.4 Proposition. If a linear subspace F of a convex space E contains a subset which is not meager-like in E, then F is Baire-like. Proof. Suppose F = U B with each B a balanced con• n n n vex rare subset of F. Each B is rare in E, therefore F n is meager-like in E, and each subset of F is meager-like in E.// For descriptive convenience^ we shall call a translate of a linear subspace a flat set , and note that a flat set contains all the points of a line if it contains two distinct points of the line. 3.5 Proposition. A convex space is Baire-like if and only if it is not meager-like. Proof. Suppose E is a convex space. If E is not meagerlike, then clearly, E is Baire-like. Conversely, suppose E = U (x + B ) where each B is balanced and convex. Now n n n n

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67 E = 1) IT>B , for otherwise, suppose x is in E but not in ^ran n^ U sp(B ) = U ™b . If sp(x) meets x + B at two disn n mn n n n tinct points, then the flat set x^ + sp(B^) contains sp(x) and, therefore, contains 0, and so is the linear subspace sp(B^). Thus x is in x^ + sp(B^) = sp(B^), a contradiction. Therefore sp(x) meets x^ + B^ in at most one point, and so the uncountable set sp(x) meets II (x + B ) = E in a countable set, which contradicts the n n n choice of x. Now, with (Bj^) increasing, each mB^ is contained in kB^, for k larger than m and n. Thus E = U^^^^n Since E is Baire-like, and (nB^) is an increasing sequence of balanced convex sets, some nB^ is not rare. Hence X + B is not rare, and so E is not meager-like.// n n 3.6 Proposition. If a subset A of a Baire-like space is absorbing at some point, then A is not meager-like. Proof. By proposition 2, we may assume A is absorbing. 3 y proposition 3, we need only show that the balanced absorbing core of A is not meager— like. Thus we may assume A is balanced and absorbing. Supposing A C ^ each B^ is balanced and convex, and x is in “ A\U^sp (B^^) . Just as in the previous proof, we may show that sp(x) meets (x^ + B^) in a countable set. Hence we may choose \a\ s 1 so that

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68 ax is not in • Since A is balanced, ax is in A C U (x + B ) , a contradiction. Thus A C u„sp(B ). Now A absorbs each point of E, so each point of E is in some sp(B^). Thus E = U^sp(B^) = = Uj,hB^. Since E is Baire-like, some kB^ is not rare, and so x^ + B^ is not rare. Therefore A is not meager-like.// For the unordered cases, a shorter proof follows from the observation that a countable union of unordered meagerlike [quasi-meager] sets is unordered meager-like [quasimeager] . 3.7 Corollary. Each non-empty open subset of a Baire-like space is not meager-like. Proof. Each non-empty open subset of a convex space is absorbing at some point.// Section 2. The subgroup theorem and applications 3.8 The difference theorem. If a subset A of a convex space E is not meager-like, and some open set U has a meager-like symmetric difference, U A A, with A, then A A is a neighborhood in E. Proof. The open set U is non-empty, for otherwise, A = (A\U) U (U\A) = U A A, and so A is meager-like, a contradiction. Proposition 4 shows that E is Baire-like.

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69 We note that if D is meager-like, then DU (x + D) is also. For suppose D C (x^^ + B^) , define = ^2n-l ~ ^2n = ^n’ ^2n-l = ^ DU (X + D) c u^(x^ + B^) U [X + U„(Xj^ + 1 = = U„(y2n + tU„(x + x^ + C2^_^)l =U^(y„ + C^), and (C ) is increasing when (B ) is. Now [U n (x + U)]\[A n (x + A) ] = = [U 0 (x + U) \A] U [U n {x + U) \(x + A) ] C (U\A) U [ (x + U) \ (x + A) ] = (U\A) U [X + (U\A) ] C (U A A) U (x + (U A A) ) . With D = U A A above, we see that the last set is meager-like. Therefore, by proposition 3, [U 0 (x + U)]\[A n (x + A) ] is meager-like. If U meets x + U, then U n (x + U) is not meager-like by corollary 7, hence A must also meet x + A. Thus U U = {x e E: U n (x + U) f 0] is contained in {x e E: A n (x + A) f 0] A A. Since U is a non-empty open set, U U is a neighborhood, and so too is A A.// 3.9 The subgroup theorem. If an additive subgroup G of a convex space E has a subset A which is not meager— like in E, and some open set U has a meager-like symmetric difference, U A A, with A, then G = E. Thus, if G is any proper subgroup of E, then G is meager-like or for no open set U is U A G meager-like.

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70 Proof. By the difference theorem U = (A A) is a neighborhood. Since G is an additive group, G D G G D A A D U. Thus G D U nU = E.// n 3.10 Corollary. If a subset A of a Baire-like space E is absorbing at some point, and some open set U has a meagerlike symmetric difference, U A A, with A, then A A is a neighborhood. Proof. By proposition 6, A is not meager-like, so the difference theorem applies.// 3.11 Corollary. Let f be an additive mapping from a convex space E into a convex space F. If, for each neighborhood V in F, there is a subset A of f ^[V] and an open set W of E such that A is not meager-like and W A A is meager-like, then f is continuous. Dually, if, for each neighborhood U in E, there is a subset A of f[U] and an open set W of F such that A is not meager-like and W A A is meager-like, then f is an open mapping onto F. Proof. For a neighborhood V* in F there is a neighborhood V with V V C V . For some subset A of f ^ [V ] and some open set W of E, A is not meager-like and W A A is meager-like. By the difference theorem, A A is a neighborhood in E.

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71 NoWj by additivity of f, we have A-ACf [V]-f [V] = f“l[V V] C [V ] • Thus f ^[V] is a neighborhood in E, and so f is continuous. Dually, the additivity of f gives f[E] f[E] = f[E E] = f[E], so f[E] is an additive subgroup of F. Since E is a neighborhood in E, there is a subset A of f [E] and an open set W of F with A not meager-like in F and A A W meagerlike. Thus f[E] = F by the subgroup theorem. For a neighborhood U' in E, there is a neighborhood U in E with U U C U' . For some subset A of f [U] and some open set W of F, A is not meager-like and W A A is meagerlike. By the difference theorem A A is a neighborhood in F. Now, by additivity of f, we have A A C f [U] f[U] = f[U U] C f[U’]. Thus f[U‘] is a neighborhood in F, and so f is an open mapping onto F.// 3.12 Corollary. Let f be an additive mapping frcm a convex space E into a convex space F. If E is Baire-like, then f is continuous if and only if for each open neighborhood V in F, f~^[V] is absorbing and for some open set W of E, W A f~^[V] is meager-like. Dually, if F is Baire-like, then f is an open mapping onto F if and only if for each open neighborhood U in E, f[U] is absorbing and for some open set W of F, W A f[U] is meager-like.

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72 Proof. Let V be an open neighborhood in F . If f is continuous, then f“^ [V] is an open neighborhood in E, and so is absorbing. With W= f ^[V], WA f ^[V] =0 is raeager-like . Conversely, f”^ [V] is absorbing in the Baire-like space E, so by property 6, f ^[V] is not meager-like. For some open set W of E, W A f~^ [V] is meager-like. With A = f"^ [V] , corollary 11 applies. Dually, let U be an open neighborhood in E. If f is an open mapping f[U] is an open neighborhood in F, and so is absorbing. Moreover, with W = f[U] , the set WA f[U] =0 is meager-like. Conversely, f[U] is absorbing in the Baire-like space F, and so is not meager-like. Since W A f[U] is meagerlike for some open set W of F, corollary 11 applies.// 3.13 Corollary. Let f be an additive mapping from a convex space E into a convex space F. If f[A] is bounded for some subset A of E such that A is not meager-like and W A A is meager-like for some open set W of E, then f is continuous. Dually, if for some bounded set A of E, f [A] is not meager-like and W A f[A] is meager-like for some open set W of F,then f is an open mapping onto F. Proof. Let V be a neighborhood in F. For some integer n, f[A] C nV. By additivity of f, A C f ^[nV] = nf ^[V].

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73 Thus f~^[V] contains A/n which is not meager-like, yet for some open set W of E, W A A and so (W/n) A (A/n) is meager-like. Thus corollary 11 applies, and f is a continuous mapping. Dually, suppose U is a neighborhood in E. For some integer n, A C nU, and so by additivity of f, f[A] C f[nU] = nf[U]. Thus f[A]/n C f[U]. Now f[A]/n is not meager-like, yet for some open set W of F, W A f[A] and so (W/n) A (f[A]/n) is meager-like. Hence corollary 11 applies, and f is an open mapping onto F.// Section 3. Analogs of the condition of Baire A set A of a topological space X is said to satisfy the condition of Baire if there is an open set U whose symmetric difference, U A A, with A is meager. Such sets are discussed in Kelley and Namioka [1], and the analog of these sets has been used repeatedly in the previous section. The collection G of all subsets of X satisfying the condition of Baire contains X and is closed under countable unions and differences, that is, C is a a-algebra. If U is open in X, then U A U = 0 is meager, and so U is in G. Hence G contains all open sets, and so, in particular, each Borel set satisfies the condition of Baire. The analogous results do not hold even for unordered meager-like. 2 . 3.14 Example. Let A be the unit circle in E = R . A is closed and so is a Borel set, yet for no open set U is

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74 U A A unordered raeager-like. For, if U is a non-empty open set, the U A A contains the non-empty open set U\A which is non-meager. For U A A to be unordered meagerlike, U A A must be meager, and so U must be empty. However, no countable collection of line segments covers A since each line segment meets A in at most two points and A is uncountable.// The class analogous to Q for unordered meager-like or quasi-meager is, however, closed under countable unions and finite intersections and contains all open sets. In the following, unordered meager-like may be replaced by unordered quasi-meager. 3.15 Proposition. Let G be the collection of all subsets A of a convex space with U A A unordered meager-like for some open set U. G contains all open sets and is closed under countable unions and finite intersections. Proof. For each open set U, U A U = 0 is unordered meager-like, and so U is in G. Suppose (A^) is a sequence in G and (U^) is a sequence of open sets with each A A^ unordered meager-like. Let A = U^A^ and U = U ^ unordered meager-like. Thus U^A^ is in G. Also (U^ n U2) A (A^ n A2) = [ (U^nU2) \ (A^nA2) ] U [ (A^HA2) \ (U^nU2) ] c [ U (U2\A2) ] u [ (A^\Up u (A2\U2) J = A A^) U (U2 A A2) ,

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75 which is unordered meager-like. Thus n and by induction G is closed under finite intersections.// Section 4. An open question An important application of the Banach subgroup theorem provides the following: If (f^) is a sequence of continuous linear mappings of a linear topological space E into a pseudo-metrizable linear topological space, then the set of points X for which Cauchy is either meager or identical with E. The proof of this result uses the closure under countable intersections of the family of sets satisfying the condition of Baire. An analog of the result is not known to be true.

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CHAPTER 4 Applications In this final chapter, we shall consider a number of applications of theorems and concepts developed above. In particular, we shall consider refinements of a categoryresult on continuous linear images of Ptak spaces as well as a modification of the Robertson and Robertson [1] closed graph and open mapping theorems. Section 1. Initial open mapping and closed graph theorems In the following, we shall depend heavily on properties of Ptak spaces discussed in Horvdth [1] . 4.1 Definition. A convex Hausdorff space F is a Ptdk space if and only if each continuous linear mapping from F into a convex Hausdorff space is open if it is almost open. Examples of Ptdk spaces are any Frechet space and the dual of any reflexive Frechet space under the strong topology. We shall use the following closed graph and open mapping theorems proved in Horvath [1]. We record the first without proof. The second follov/s from proposition 2.12 and the above definition. 76

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77 4.2 Theorem [Robertson and Robertson]. If f is a linear mapping from a barrelled Hausdorff space E into a Ptdk space F which is closed in E x F, then f is continuous. 4.3 Theorem [Ptdk] . If g is a continuous linear mapping from a pt^ space F onto a barrelled Hausdorff space E, then g is an open mapping. Section 2. Extensions of a theorem of Banach Banach [1, p. 38] showed that the continuous linear image of a complete metrizable linear space in a space of the same kind is either meager or the whole space. Robertson and Robertson [1] extended this result in convex spaces. We refine their results by specifying in what sense the image is meager. 4.4 Proposition. If g is a continuous linear mapping from a Ptak space F into a convex Hausdorff space E, then g is a surjection if g[E] is both dense and barrelled. Proof. Since E is Hausdorff, we need only show that g [F] is complete in order to show g [F ] is closed. From this, g [F] = g [F] = E. Since g[F] is barrelled and Hausdorff, g is an open mapping from theorem 3. Let n be the canonical mapping from F onto the quotient space H = F/N(g), and g be the

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78 one-to-one linear mapping from H into E induced by g, so that g = g o r\. Since g is open and continuous, g is also. Therefore g[F] = g[H] is the embedding of H in E by g. Now E is Hausdorff and g is continuous, so N(g) is a closed linear subspace of the Ptak space F. Thus H = F/N(g) is a Pt^ space, so g[F] is as well. Yet a Ptdk space is complete, thus g[F] is complete.// Clearly a non-dense linear subspace is meager. The remark following proposition 2.9 shows that a linear subspace which is not barrelled is meager. Hence the above proposition includes Banach's result for Frechet spaces. As an application of the proposition, we have the following . 4.5 Proposition. Let {v^:F^ ^ countable family of linear mappings v^ from the Pt^k spaces F^ into the linear space F which is covered by If F is given a convex Hausdorff topology for which each V is continuous, then some v is a surjection or F is n n not unordered Baire-like. Proof. Suppose each From proposition 4, each V [Fj^] is rare or not barrelled. Yet F is covered by fv FF 11 . and so F is not unordered Baire-like by proposin ' n^ n^ tion 2.9.//

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79 An Gxainple of S. A. Saxon shows ths above statement to be false when "unordered" is omitted.* This proposition may be applied to the space E described in example 2,8 to show immediately that E is not unordered Baire-like. The following is a further refinement of a category result of Robertson and Robertson [1]. 4.6 Proposition. Let ^^neuj ^ countable family of linear mappings from the Pt^k spaces into the linear space F which is covered by ^ have a convex topology for which each v^ is continuous. If g is a continuous linear mapping from F into a convex Hausdorff space E, then g is a surjection or g[F] is unordered meager-like. Proof. Suppose g[F] is not unordered meager-like. Clearly, g[F] is dense. By the unordered version of proposition 3.4, g[F] is unordered Baire-like. Now g[F] = U qfv fF 1], and so some g [v [F ]] is dense in g[F] and n n n n n barrelled by proposition 2.9. Since g[F] is dense in E, qrv \F 11 is dense in E. Proposition 4 applies for the continuous linear mapping g o v^. Hence g ° v^, and so g, is a sur j action.// * S. A. Saxon, personal communication.

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80 Section 3. The Robertson and Robertson theorems We now state and prove the Robertson and Robertson closed graph theorem with convex Baire replaced by unordered Baire-lihe. The essential modification in the proof occurs in the second line^ the rest of the proof is explicated for completeness . 4.7 Theorem [Robertson and Robertson]. Let a family [u. :E^ -» ^ countable family "* the linear mappings u^ and v^ from the Hausdorff unordered Baire-like spaces and the Ptak spaces into the linear spaces E and F respectively. Suppose E has the strongest convex topology for which each u^ is continuous, and F has a convex Hausdorff topology for which each v^ is continuous. If F is covered by then each linear mapping f from E into F which is closed in E x F is continuous. Proof. Suppose E is a Hausdorff unordered Baire-like is dense in E and barrelled by proposition 2.9. We obtain the following diagram space. Now E = f ^[F] = U^f ^ fv [F ]], so some L=f ^ [v„ [F ] ] ^n n ^ nn V = V n F -g ^ F L g

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81 where n is the canonical mapping from onto the quotient space M, and v is the continuous linear mapping from M into F induced by v = so that v = v o r| • Since v is one-to-one and onto v[F^] = v[M] D f[L], we have g = v ^ o f^^ a well-defined linear mapping from L into M. Let h be the continuous mapping from L x M to E x F defined by h(x,y) = (x,v(y)). Identifying f and g with their graphs, we have g = h ^[f] and, since f is closed in E X F, g is closed in L x M. Now F is Hausdorff, so N(v) is a closed linear subspace of the Ptak space F. Therefore M = F^/N(v) is a Ptak space. Also L is a barrelled Hausdorff space. By theorem 2, g is continuous . A Ptak space is complete and Hausdorff, and so, as L is dense in E, g has a unique continuous extension g from E into the Ptak space M. For x in L, v (g (x) ) = v (v f (x) ) = f(x). If this holds for all x in E, then f = vo g, and so f is a composition of continuous functions. Assume x' is in E with f(x') 7 ^ v(g(x')). Since f is closed in E X F, there are neighborhoods U in E and V in F with (x' + U) X (v(g(x')) + V) disjoint from f. By continuity of Vo g, there is a neighborhood U' in E with U'CU and Vo g[U'] C V. Since L is dense in E, there is an element x in L H (x' + U' ) . Now f contains (x,f(x)), and (x,f(x)) = (x,vo g (x)) e (x‘ +U') x ( v o g [x* + U' ] ) C (x' + U) X (v o g (x‘ ) + V) , a contradiction. Thus vo g (x) = f (x) for all x in E.

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82 For the general case, we need to show that each mapping f. = f o u. from E. into F is continuous. Fix i and let h be the continuous mapping from x F into E X F defined by h(x,y) = (u^(x),y). Clearly, ^ ^ and, since f is closed in E x F, the mapping f^ is closed in E. X F. The initial portion of the proof shows that f. is continuous.// There is the corresponding open mapping theorem which we record without proof. 4.8 Theorem. Let the hypotheses of theorem 7 hold. If F is covered by then each continuous linear mapping g from F onto E is an open mapping. Valdivia [2] has studied a closed graph theorem which includes the Robertson and Robertson closed graph theorem. In place of Pt^k spaces, he uses a type of space characterized by a closed graph property which is satisfied by Ptak spaces. In place of convex Baire spaces, Valdivia defines and uses infra-Baire spaces. 4.9 Definition. A convex space E is an infra-Baire space if and only if there is a convex Baire space G such that E is embedded in G as a linear subspace of finite codimension. Since a convex Baire space has the unordered Bairelike property, which is inherited by linear subspaces of

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83 countable codimension, an infra-Baire space is unordered Baire-like. Clearly, a space which distinguishes between infra-Baire and the convex Baire properties answers in the negative the question of inheritance of the Baire property by linear subspaces of countable codiraension. Although Valdivia [2] proves several permanence prop 02 ^t,ies for infra-Baire spaces, which are shared by unorder ed Baire-like spaces, productivity of infra-Baire spaces is left an open question. It may be seen that the infra-Baire property is inherited by linear subspaces of countable codimension. The productivity and inheritance properties for unordered Baire-like spaces shows that un— ordered Baire-like in theorems 7 and 8 may be replaced by products of countable-codimensional subspaces of convex Baire spaces or, equivalently, by products of infra-Baire spaces .

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BIBLIOGRAPHY I. Amemiya and Y. Koraura [1] "Uber nicht-vollstandige Montelraurae , " Math. Ann. 177 (1968) ^ 21 'I -211 . S. Banach [ 1 ] Thdorie des Operations Lindaires , Warsaw, 1932. N. Bourbaki [ 1 ] Espaces Vectoriels Topologiques , Chapitres III-V, Livre V, Actualites Scientif iques et Indus trielles 1229, Hermann, Paris, 1955. [2] Elements of Mathematics, General Topology , Part 2, Hermann, Paris, in translation, AddisonWesley, Reading, Mass., 1966. H. F. Cullen [ 1 ] Introduction to General Topology , D. C. Heath, Boston, 1968. M. M. Day [ 1 ] Normed Linear Spaces , Ergebnesse der Mathematik und ihrer Grenzgebiete , n. F., Heft 21, Springer, Berlin, 1962. J. De Groot [1] "Subcompactness and the Baire category theorem," Nederl. Akad. Wetensch. Indag. Math. 25 (1963), 761-767. 84

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85 M. De Wilde and C. Houet [1] "On increasing sequences of absolutely convex sets in locally convex spaces," Math. Ann. 192 (1971), 257-261. Z. Frolik [1] "Baire spaces and some generalizations of complete metric spaces," Czech. Math. J. 11, 86 (1961), 237-247. [2] "Remarks concerning the invariance of Baire spaces under mappings," Czech. Math. J. 11, 86 (1961), 381-384. J. Horvath [1] Topological Vector Spaces and Distributions , Vol. 1, Addison-Wesley, Reading, Mass., 1966. T. Husain [ 1 ] The Open Mapping and Closed Graph Theorems in Topological Vector Spaces , Oxford University Press, London, 1965. J. L. Kelley [ 1 ] General Topology , D. Van Nostrand, Princeton, New Jersey, 1955. J. L. Kelley and I. Namioka [1] Linear Topological Spaces , D. Van Nostrand, Princeton, New Jersey, 1963. M. Levin and S. A. Saxon [1] "A note on the inheritance of properties of locally convex spaces by subspaces of countable codimension," Proc. Amer. Math. Soc. 29 (1971), 97-102.

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86 J. C. Oxtoby [1] "Cartesian products of Baire spaces," Fund. Math. 49 (1961), 157-166. A. P. Robertson and W. J. Robertson [1] "On the closed graph theorem," Proc. Glasgow Math. Assoc. 3 (1956) , 9-12. [2] Topological Vector Spaces , Cambridge Tracts in Mathematics and Mathematical Physics 52, Cambridge University Press, Cambridge, 1964. S. A. Saxon [1] "Nuclear and product spaces, Baire-like spaces, and the strongest locally convex topology," Math. Ann., to appear. [2] " (LF) -spaces , quasi-Baire spaces, and the strongest locally convex topology," to appear. S. A. Saxon and M. Levin [1] "Every countable-codimensional subspace of a barrelled space is barrelled," Proc. Amer . Math. Soc. 29 (1971), 91-96. Valdivia [1] "Absolutely convex Ann. Inst. Fourier 3-13. sets in barrelled spaces," (Grenoble) 21, 2 (1971) , [2] Sobre el Teorema de la Grafica Cerrada , Seminario Mathematico de Barcelona, Collectanea Mathematics, Vol. 22, Fasc. 1° , Grafica Elzeviriana, S. A., Barcelona, 1971.

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BIOGRAPHICAL SKETCH Aaron Rodwell Todd was born December 25, 1942, in El Portal, Florida to Caroline Osborne Hall and Edmund Neville Todd. After graduating from Miami Edison Senior High School in 1960 and the University of Michigan in 1964, he taught mathematics for three years in Ghana. Janet Margaret Dakin and he were married December 21, 1966 in Llanrhaerdr, Wales. They both attended the University of Leeds for a year beginning October, 1967. Julian Garfield Todd was born to them October 2, 1968, in Sutton Coldfield, England. All three arrived January, 1969, in Gainesville, Florida where Julian enjoyed his most formative years while his parents taught mathematics and English literature and attended the University of Florida. 87

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I certify that I have read this study and that in niy opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Dcotor of Philosophy. j.^K. ferboks. Chairman Professor of Mathematics I certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philosophy. S. A. Saxon Assistant Professor of Mathematics I certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philosophy. y5 n'OAA'c hovic / Z. Pop-Stojahovic / Professor of Mathematics I certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philosophy. A. K. Varma Associate Professor of Mathematics

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I certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philosophy. I certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philosophy. This dissertation was submitted to the Department of Mathematics in the College of Arts and Sciences, and to the Graduate Council, and was accepted as partial fulfillment of the requirements for the degree of Doctor of Philosophy. A. R. Bednarek Professor of Mathematics Ward Hellstrom Professor of English Dean, Graduate School August, 1972