A property of locally convex Baire spaces

Todd, Aaron R.; Saxon, Stephen A.

doi:10.1007/bf01431526

Cited by 55 publications

(31 citation statements)

References 11 publications

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“…(Mazur and Orlicz (1948), p. 208). Although the complete space E = \\ I E i need not be metrizable, the result of the corollary may also be obtained for this choice of E h iel, from the fact that a product of complete metrizable spaces is Baire (Bourbaki (1948), p. 4, Exercise 7), and such subspaces F of E, as in the corollary, are Baire if E is Baire (Todd and Saxon (1973), p. 32, Theorem 4.11).…”

Section: Corollary If E Is the Product Of Linear Topological Spaces mentioning

confidence: 96%

“…As a direct corollary of Theorem 4.1 of Todd and Saxon (1973), the following requires no topology and is true without the requirement that 38 consist of closed sets; however, the present formulation is adequate for its use in Theorem 3.1. PROOF.…”

Section: Corollary the Product Of Barrelled Spaces Is Barrelledmentioning

confidence: 99%

“…Suppose 88 = {5 n } nS 1 is a family of closed balanced convex subsets of E which covers F. It is readily seen (Todd and Saxon (1973), p. 31, Lemma 4.9) that!% covers E. As no factor E t may be covered by 38i -{kB: k is an integer, and B is a member of J 1 which is not absorbing in E t }, Theorem 3.1 implies that there is an element B of 3D which is absorbing in E.…”

Section: Corollary If E Is the Product Of Linear Topological Spaces mentioning

confidence: 99%

“…If locally convex linear topological spaces alone are considered, and the cover contains integral multiples of a single balanced convex rare set, we obtain a characterization of barrelled spaces (those locally convex linear topological spaces all of whose closed absorbing balanced convex subsets (barrels) are neighborhoods of the origin). Other restrictions on the covers, also of an algebraic nature, result in properties intermediate to the Baire and barrelled properties (see Todd and Saxon (1973) for a detailed discussion of these properties and citations of other papers by Saxon). In particular, a locally 282 Aaron R. Todd [2] convex linear topological space is unordered Baire-like if and only if it has no cover by a countable family of rare balanced convex subsets, equivalently, each cover of the space by a countable family of closed balanced convex sets contains a neighborhood of the origin.…”

Section: Introductionmentioning

confidence: 99%

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Coverings of products of linear topological spaces

Todd

1980

J Aust Math Soc A

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An extension of the Banach-Mackey theorem is used to prove a theorem about countable families of closed balanced convex sets that cover a product of linear topological spaces. This theorem clarifies proofs that certain Baire-type properties, including the unordered Baire-like property, are preserved under products. A modification of the theorem is used to show that a property involving the bounded-absorbing sequences of DeWilde and Houet is also productive. Finally, a question is posed about balanced absorbing sets relating to products of linear Baire spaces.

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Section: Corollary If E Is the Product Of Linear Topological Spaces mentioning

confidence: 96%

Section: Corollary the Product Of Barrelled Spaces Is Barrelledmentioning

confidence: 99%

Section: Corollary If E Is the Product Of Linear Topological Spaces mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Coverings of products of linear topological spaces

Todd

1980

J Aust Math Soc A

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“…A l.c.s. E is said to be unordered Baire-like (UBL in brief) [12] if, given a sequence of closed absolutely convex subsets of E covering E, one of them is a neighbourhood of the origin in E. A l.c.s. E is said to be Baire-like (BL in brief) [11] if, given an increasing sequence of closed absolutely convex subsets of E covering E, then one of them is a neighbourhood of the origin in E. One has that…”

Section: Preliminariesmentioning

confidence: 99%

On some spaces of vector-valued bounded functions

Ferrando¹

1999

MATH. SCAND.

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Assuming AE is an algebra of subsets of a non-empty set and X is a normed space, I investigate whether or not certain barrelledness conditions, some of them introduced in the seventies by Saxon and Valdivia, are enjoyed by several subspaces of the linear space of all those bounded X -valued functions defined on which are the uniform limit of a sequence of X-valued AE-simple functions equipped with the supremum-norm. PreliminariesIn what follows will be a non-empty set, AE an algebra of subsets of and X a normed space. By I 0 AEY X we denote the linear space over the field K of the real or complex numbers of all X -valued AE-simple functions defined on . If X K, we will write I 0 AE instead of I 0 AEY K . We represent by B AEY X , or by B Y X , the linear space over K of all bounded X -valued functions defined on which are the uniform limit of a sequence in I 0 AEY X . We will assume that B AEY X is equipped with the supremum-norm f k k sup f 3 k kX 3 P f gand will consider I 0 AEY X as a subspace of B AEY X . If AE is not a '-algebra, then f À1 x , for f P B AEY X and x P X , need not be an element of AE, although certainly it is an element of the '-algebra generated by AE. The subspace of B AEY X formed by all countablyvalued functions will be denoted by K AEY X or by K Y X , while K 0 AEY X will stand for the linear subspace of K AEY X consisting of all those countably-valued f for which there exists a countable partition A n Y n P N f gof by elements of AE such that f is constant in each set A n , n P N. If f P K AEY X is such that f À1 x P AE for all x P X , then clearly f P K 0 AEY X . Naturally K 0 AEY X coincides with K AEY X whenever AE is a '-algebra, although in general K 0 AEY X is a dense subspace of K AEY X , since I 0 AEY X K 0 AEY X . When X K we will write K 0 AE instead of K 0 AEY K . If E we will represent by B EY X the subspace of B AEY X of all those functions f whose

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On the Mathematical Work of Professor Manuel López-Pellicer

Ferrando

2019

Descriptive Topology and Functional Analysis II

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A property of locally convex Baire spaces

Cited by 55 publications

References 11 publications

Coverings of products of linear topological spaces

Coverings of products of linear topological spaces

On some spaces of vector-valued bounded functions

On the Mathematical Work of Professor Manuel López-Pellicer

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