Networks for the weak topology of Banach and Fréchet spaces

Gabriyelyan, Saak; Ka̧kol, Jerzy; Kubiś, Wiesław; Marciszewski, Witold

doi:10.1016/j.jmaa.2015.07.037

Cited by 16 publications

(17 citation statements)

References 20 publications

(39 reference statements)

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Theorem 3.5 and Corollary 3.7 immediately imply the following result which might be also extracted from the proof of [31,Corollary 2.18] and is proved in [12].…”

Section: Nagata-smirnov Metrization Theorem Impliesmentioning

confidence: 55%

“…Let us note that in the following theorem the implication (i) ⇔ (ii) generalizes [4, Theorem 1.9], the implication (i) ⇔ (iii) is proved in [12] and (i) ⇔ (v) follows from [24].…”

Section: Metrizability Conditions For Topological Spacesmentioning

confidence: 82%

“…In [12] it is shown that a topological space X is cosmic (resp. an ℵ 0 -space) if and only if X is a Lindelöf σ-space (resp.…”

Section: Proposition 311 Each P-space Is a Strict ℵ-Space And Eachmentioning

confidence: 99%

“…Then Theorem 5.1.27 of [8] implies that X = ⊕ i∈I X i , where X i is a clopen Lindelöf subset of X for every i ∈ I. By [12], any X i is a locally compact cosmic space. Hence X i is a separable metrizable space by [8, 3.3.5].…”

Section: Metrizability Conditions For Topological Spacesmentioning

confidence: 99%

See 3 more Smart Citations

On P-spaces and related concepts

Gabriyelyan

Ka̧kol

2015

Topology and its Applications

Self Cite

View full text Add to dashboard Cite

The concept of the strong Pytkeev property, recently introduced by Tsaban and Zdomskyy in [32], was successfully applied to the study of the space C c (X) of all continuous real-valued functions with the compact-open topology on some classes of topological spaces X including Čech-complete Lindelöf spaces. Being motivated also by several results providing various concepts of networks we introduce the class of P-spaces strictly included in the class of ℵ-spaces. This class of generalized metric spaces is closed under taking subspaces, topological sums and countable products and any space from this class has countable tightness. Every P-space X has the strong Pytkeev property. The main result of the present paper states that if X is an ℵ 0 -space and Y is a P-space, then the function space C c (X, Y ) has the strong Pytkeev property. This implies that for a separable metrizable space X and a metrizable topological group G the space C c (X, G) is metrizable if and only if it is Fréchet-Urysohn. We show that a locally precompact group G is a P-space if and only if G is metrizable.

show abstract

“…Theorem 3.5 and Corollary 3.7 immediately imply the following result which might be also extracted from the proof of [31,Corollary 2.18] and is proved in [12].…”

Section: Nagata-smirnov Metrization Theorem Impliesmentioning

confidence: 55%

“…Let us note that in the following theorem the implication (i) ⇔ (ii) generalizes [4, Theorem 1.9], the implication (i) ⇔ (iii) is proved in [12] and (i) ⇔ (v) follows from [24].…”

Section: Metrizability Conditions For Topological Spacesmentioning

confidence: 82%

“…In [12] it is shown that a topological space X is cosmic (resp. an ℵ 0 -space) if and only if X is a Lindelöf σ-space (resp.…”

Section: Proposition 311 Each P-space Is a Strict ℵ-Space And Eachmentioning

confidence: 99%

Section: Metrizability Conditions For Topological Spacesmentioning

confidence: 99%

See 2 more Smart Citations

On P-spaces and related concepts

Gabriyelyan

Ka̧kol

2015

Topology and its Applications

Self Cite

View full text Add to dashboard Cite

show abstract

“…In it is proved that if E is a Banach space whose strong dual is separable, then

E_{w}

is an ℵ 0 ‐space. In [, Corollary 5.6] it was shown that a Banach space which does not contain an isomorphic copy of ℓ 1 has separable dual if and only if

E_{w}

is an ℵ 0 ‐space. Next theorem extends this result to quasi‐

(D F)

‐spaces.…”

Section: Quasi‐(df)‐spacesmentioning

confidence: 99%

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

Ferrando

Gabriyelyan

Ka̧kol

2019

Mathematische Nachrichten

View full text Add to dashboard Cite

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.

show abstract

Separable (and Metrizable) Infinite Dimensional Quotients of $$C_p(X)$$ and $$C_c(X)$$ Spaces

Ka̧kol

2019

Descriptive Topology and Functional Analysis II

View full text Add to dashboard Cite

Networks for the weak topology of Banach and Fréchet spaces

Cited by 16 publications

References 20 publications

On P-spaces and related concepts

On P-spaces and related concepts

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

Separable (and Metrizable) Infinite Dimensional Quotients of $$C_p(X)$$ and $$C_c(X)$$ Spaces

Contact Info

Product

Resources

About