On nonbornological barrelled spaces

Valdivia, Manuel

doi:10.5802/aif.410

Cited by 14 publications

(9 citation statements)

References 5 publications

(5 reference statements)

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“…Using a result of Noble [17, p. 270] one can show that G is holomorphically quasibarrelled (see [19, Remark 17], and [33,Chapter 12]). It is proved in [37] 2. Sets of holomorphic mappings on spaces of continuous functions.…”

Section: Introductionmentioning

confidence: 90%

Locally Bounded Sets of Holomorphic Mappings

Bonet¹,

Galindo²,

Garcı́a³

et al. 1988

Transactions of the American Mathematical Society

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ABSTRACT. Several results and examples about locally bounded sets of holomorphic mappings defined on certain classes of locally convex spaces (Baire spaces, (DF)-spaces, C(X)-spaces) are presented. Their relation with the classification of locally convex spaces according to holomorphic analogues of barrelled and bornological properties of the linear theory is considered. Introduction.The main topic discussed in this article is the classification of locally convex spaces according to the holomorphic analogues of barrelled, bornological and quasibarrelled properties introduced for the linear theory. This question is related to the study of locally bounded sets of holomorphic mappings defined on open subsets of locally convex spaces.A satisfactory classification can only be obtained in the case of special classes of locally convex spaces, e.g., metrizable spaces, Baire spaces, (DF)-spaces and spaces of type C(X). Some positive results are presented and several examples will show the essential differences between the linear and the holomorphic theory.We shall use standard notations of locally convex spaces as in [21 and 22] and of infinite holomorphy as in [15 and 33]. The word "space" will mean separated locally convex topological vector space. If E is a space, cs(.E) denotes the set of all continuous seminorms on E. A family X of mappings defined on an open subset U of a space E with values in a normed space F is called locally bounded if for every x E U there is a neighbourhood V of x contained in U and M > 0 with ||/(j/)|| < M

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Section: Introductionmentioning

confidence: 90%

Locally Bounded Sets of Holomorphic Mappings

Bonet¹,

Galindo²,

Garcı́a³

et al. 1988

Transactions of the American Mathematical Society

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“…A l.c.s. E is said to be totally barrelled [15] (TB for short) if, given a sequence E n of subspaces of E covering E, there is one of them which is barrelled and its closure is of finite codimension in E. If E is metrizable, this is equivalent (see 9X3X3 of [10]) to saying that there is some E m which is barrelled. A l.c.s.…”

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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“…So, if E is not countably quasi-suprabarrelled, there exists an increasing sequence of barrelled subspaces of E covering E, {F n : n £ N}, such that codg F n > N o for every n £ N. Then {F n : n £ N and F n D E(I \ J n ) with J n a finite subset of / } is also an increasing sequence of barrelled subspaces of E covering E, [12,Proposition 4]. Now, by Lemma 2, there exists some j £ I such that {F n : n £ N and cod.E({j}) F n n E({j}) > No} covers E({j}), which is not possible since E({j}) is countably quasi-suprabarrelled.…”

Section: Theorem 1 If {Ei: I £ 1} Is a Family Of Countably Quasi-supmentioning

confidence: 99%

“…Let us recall a space E is quasi-suprabarrelled [1] if, given an increasing sequence of subspaces of E covering E, there is one which is barrelled; E satisfies condition (G) [4] if, given a sequence of subspaces of E covering E, there is one which is barrelled; E is quasitotally barrelled [2] if, given a sequence of subspaces of E covering E, there is one which is barrelled and its closure has countable codimension in E; E is totally barrelled [12] if, given a sequence of subspaces of E covering E, there is one which is barrelled and its closure has finite codimension in E; E is unordered Baire-like [6] if, given a sequence of closed absolutely convex subsets of E covering E, there is one which is a neighbourhood of the origin; and E is suprabarrelled [9] ((bd) in [5]) if, given an increasing sequence of subspaces of E covering E, there is one which is barrelled and dense in E. The relationship among these classes of spaces is the following: In this paper we shall introduce a class of spaces located between quasi-totally barrelled spaces and quasi-suprabarrelled spaces, which enjoys good permanence properties, and satisfies a closed graph theorem.…”

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confidence: 99%