On compactness in locally convex spaces

Cascales, B.; Orihuela, J.

doi:10.1007/bf01161762

Cited by 98 publications

(117 citation statements)

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“…Thus, if G is separable, the equivalence follows from Theorem 2.9. Under the assumption of ordinary reflexivity (without separability), taking into account [6,Theorem 11 (ii)], the proof follows from Propositions 2.8 and 2.4(c).…”

Section: Proposition 28 Let G Be An Abelian Topological Group With mentioning

confidence: 74%

A property of Dunford-Pettis type in topological groups

Martín-Peinador

Tarieladze

2003

Proc. Amer. Math. Soc.

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Abstract. The property of Dunford-Pettis for a locally convex space was introduced by Grothendieck in 1953. Since then it has been intensively studied, with especial emphasis in the framework of Banach space theory.In this paper we define the Bohr sequential continuity property (BSCP) for a topological Abelian group. This notion could be the analogue to the DunfordPettis property in the context of groups. We have picked this name because the Bohr topology of the group and of the dual group plays an important role in the definition. We relate the BSCP with the Schur property, which also admits a natural formulation for Abelian topological groups, and we prove that they are equivalent within the class of separable metrizable locally quasi-convex groups.For Banach spaces (or for metrizable locally convex spaces), considered in their additive structure, we show that the BSCP lies between the Schur and the Dunford-Pettis properties. PreliminariesThe aim of this paper is to give an analogue to the Dunford-Pettis property in the context of topological Abelian groups, as explained in Section 3. In this spirit we define in Section 2 the Bohr sequential continuity property, which holds for a class of groups larger than that of Hausdorff locally compact Abelian (LCA) groups. The most satisfactory situation would be, if for any locally convex space E, the BSCP were equivalent to the Dunford-Pettis property. The question does not go so smoothly; we have obtained some related results and an insight to properties which so far had only been treated for topological vector spaces and, in fact, they can be extended to topological Abelian groups.We now introduce notation and relevant facts. Let G be an Abelian topological group; we denote by N e (G) the set of all neighborhoods of the neutral element e. The group of all continuous homomorphisms (also named characters) from G into the torus T = {z ∈ C : |z| = 1}, considered with pointwise multiplication,

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Section: Proposition 28 Let G Be An Abelian Topological Group With mentioning

confidence: 74%

A property of Dunford-Pettis type in topological groups

Martín-Peinador

Tarieladze

2003

Proc. Amer. Math. Soc.

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“…A uniform space L is called trans-separable if every uniform cover of L admits a countable subcover ( [7], [6]). This notion has occured to be useful in the work of ( [14], [1], [15]) while studying the metrizability of precompact sets in locally convex spaces (see also [9]). Drewnowski [2] had actually coined the word "trans-separable" and it has been further used by Robertson [15].…”

Section: N(a W) = {Fe Cb(x E) : F(a) C W}mentioning

confidence: 99%

Trans-Separability in Spaces of Continuous Vector-Valued Functions

Khan¹

2004

Demonstratio Mathematica

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Abstract. Let X be a completely regular Hausdorff space and E a Hausdorff topological vector space (TVS). In this note, we study the notion of trans-separability for certain subspaces of Cb(X, E) endowed with the c-compact-open and some related topologies.Let X be a completely regular Hausdorff space and E a Hausdorff topological vector space (TVS) with a base W of balanced neighbourhoods of 0, and let Cb(X, E) (resp. C RC (X, E) ) denote the vector space of all continuous i?-valued functions / on X such that F(X) is bounded (resp. relatively compact). Let CQ(X,E) be the subspace of Cb(X,E) consisting of those functions which vanish at infinity.When E is the real or complex field, C B {X,E) = C RC (X,E) and we denote this by Cb(X) and CQ(X, E) by CQ(X). The a-compact-open topology a (resp. compact-open topology k, countable-open topology ao) [4,5,11] on CB(X,E)is defined as the linear topology which has a base of neighbourhoods of 0 consisting of all sets of the form N(A, W) = {fe Cb(X, E) : f(A) C W},where A varies over all er-compact (resp. compact, countable) subsets of X and W € W. The uniform topology u on Cb(X, E) is the linear topology which has a base of neighbourhoods of 0 consisting of all sets of the formwhere W e W. Clearly, a 0 < a < u and k < a. Further, by ([11], Theorem 3.3), a = u iff X = A for some a-compact subset A; k = a iff every cr-compact subset of X is relatively compact; oo = u iff X is separable. For any A C X, let || . m denote the seminorm on Cb(X) given by || / \\ A = sup l6j4 |/(®)|, / e C b (X). We shall denote by C b (X) ® E the vector subspace of Cb(X, E) spaned by the set of all functions of the form 1991 Mathematics Subject Classification: Primary 46E10; Secondary 46E40, 46A10.

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“…But E a := (E, a(E, E')) does not belong to class 0. Otherwise, since E a is dense in the product R 1 for some set /, the space R 7 would also belong to class (9,[6,Proposition 8]. But R 7 is a Baire space, so [7,Theorem 4.8] applies to show that R 7 has countable tightness, that is, for every AcW and every x € A there exists a countable set B C A such that x € B. Consequently, / is countable and E is finite-dimensional, a contradiction.…”

Section: Corollary 5 the Strong Dual (E' $(E\ E)) Of A Locally Conmentioning

confidence: 99%

“…Finally, recall that a locally convex space E belongs to class & if there is a family where N is endowed with the discrete and N N with the product topology, respectively, [6]. Condition (c) implies that every set A a is a(E', ^-relatively countably compact.…”

Section: Introductionmentioning

confidence: 99%