On topological properties of Fréchet locally convex spaces with the weak topology

Gabriyelyan, Saak; Ka̧kol, Jerzy; Kubzdela, Albert; Pellicer, Manuel López

doi:10.1016/j.topol.2015.05.075

Cited by 17 publications

(26 citation statements)

References 28 publications

(49 reference statements)

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“…Following [22], a locally convex space E is said to have the Rosenthal property if every bounded sequence in E has a subsequence which either (1) is Cauchy in the weak topology, or (2) is equivalent to the unit basis of ℓ 1 . Proposition 2.6.…”

Section: Proof Of Theorem 12mentioning

confidence: 99%

Locally convex spaces and Schur type properties

Gabriyelyan¹

2019

Ann. Acad. Sci. Fenn. Math.

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In the main result of the paper we extend Rosenthal's characterization of Banach spaces with the Schur property by showing that for a quasi-complete locally convex space E whose separable bounded sets are metrizable the following conditions are equivalent: (1) E has the Schur property, (2) E and Ew have the same sequentially compact sets, where Ew is the space E with the weak topology, (3) E and Ew have the same compact sets, (4) E and Ew have the same countably compact sets, (5) E and Ew have the same pseudocompact sets, (6) E and Ew have the same functionally bounded sets, (7) every bounded non-precompact sequence in E has a subsequence which is equivalent to the unit basis of ℓ1 and (8) every bounded non-precompact sequence in E has a subsequence which is discrete and C-embedded in Ew.2000 Mathematics Subject Classification. 46A03, 46E10.

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Section: Proof Of Theorem 12mentioning

confidence: 99%

Locally convex spaces and Schur type properties

Gabriyelyan¹

2019

Ann. Acad. Sci. Fenn. Math.

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“…The latter class contains all Banach spaces, as well as every (F S)-space (see [5]). In [13] we proved the following…”

Section: Lemma 51 ([18 Cor 68]) the Strong Dual Of A Lcs E Is Trmentioning

confidence: 85%

“…So the trans-separable lcs E is covered by a sequence of metrizable bounded absolutely convex sets (Q n ) n . Now Corollary 4.12 of [13] implies that E is separable. As E is separable, then E is a weakly ℵ 0 -space by Proposition 5.2(i).…”

Section: Theorem 57 Let E Be a Fréchet Lcs Not Containing A Copy Ofmentioning

confidence: 95%

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Networks for the weak topology of Banach and Fréchet spaces

Gabriyelyan

Ka̧kol

Kubiś

et al. 2015

Journal of Mathematical Analysis and Applications

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Please cite this article in press as: S. Gabriyelyan et al., Networks for the weak topology of Banach and Fréchet spaces, J. Math. Anal. Appl. (2015), http://dx.Abstract. We start the systematic study of Fréchet spaces which are ℵ-spaces in the weak topology. A topological space X is an ℵ 0 -space or an ℵ-space if X has a countable k-network or a σ-locally finite k-network, respectively. We are motivated by the following result of Corson (1966): If the space C c (X) of continuous realvalued functions on a Tychonoff space X endowed with the compact-open topology is a Banach space, then C c (X) endowed with the weak topology is an ℵ 0 -space if and only if X is countable. We extend Corson's result as follows: If the space E := C c (X) is a Fréchet lcs, then E endowed with its weak topology σ(E, E ) is an ℵ-space if and only if (E, σ(E, E )) is an ℵ 0 -space if and only if X is countable. We obtain a necessary and some sufficient conditions on a Fréchet lcs to be an ℵ-space in the weak topology. We prove that a reflexive Fréchet lcs E in the weak topology σ(E, E ) is an ℵ-space if and only if (E, σ(E, E )) is an ℵ 0 -space if and only if E is separable. We show however that the nonseparable Banach space 1 (R) with the weak topology is an ℵ-space.

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“…Proof. Since has a fundamental compact resolution by (see Fact 1 in the proof of Proposition 4.7 in [24]), the space ( ) has a -base by [16]. As is a -space, ( ) is barrelled.…”

Section: Example 43mentioning

confidence: 99%

Bounded sets structure of CpX and quasi‐(DF)‐spaces

Ferrando

Gabriyelyan

Ka̧kol

2019

Mathematische Nachrichten

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For wide classes of locally convex spaces, in particular, for the space Cpfalse(Xfalse) of continuous real‐valued functions on a Tychonoff space X equipped with the pointwise topology, we characterize the existence of a fundamental bounded resolution (i.e., an increasing family of bounded sets indexed by the irrationals which swallows the bounded sets). These facts together with some results from Grothendieck's theory of (DF)‐spaces have led us to introduce quasi‐(DF)‐spaces, a class of locally convex spaces containing (DF)‐spaces that preserves subspaces, countable direct sums and countable products. Regular (LM)‐spaces as well as their strong duals are quasi‐(DF)‐spaces. Hence the space of distributions D′false(normalΩfalse) provides a concrete example of a quasi‐(DF)‐space not being a (DF)‐space. We show that Cpfalse(Xfalse) has a fundamental bounded resolution if and only if Cpfalse(Xfalse) is a quasi‐(DF)‐space if and only if the strong dual of Cpfalse(Xfalse) is a quasi‐(DF)‐space if and only if X is countable. If X is metrizable, then Ckfalse(Xfalse) is a quasi‐(DF)‐space if and only if X is a σ‐compact Polish space.

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On topological properties of Fréchet locally convex spaces with the weak topology

Cited by 17 publications

References 28 publications

Locally convex spaces and Schur type properties

Locally convex spaces and Schur type properties

Networks for the weak topology of Banach and Fréchet spaces

Bounded sets structure of CpX and quasi‐(DF)‐spaces

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