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

Ferrando, J. C.; Gabriyelyan, Saak; Ka̧kol, Jerzy

doi:10.1002/mana.201800085

Cited by 12 publications

(11 citation statements)

References 40 publications

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“…Proof. Let X be the countable Tychonoff space defined in Example 4.10 of [9]. Then, as we proved in [9], L(X) is a quasi-(DF )-space.…”

Section: Proof Of Main Resultsmentioning

confidence: 69%

“…Following [9], an lcs (E, τ ) is called a quasi-(DF )-space if E admits a fundamental bounded resolution and belongs to the class G. For the definition of the class G, we refer the reader to the original paper [5] by Cascales and Orihuela or to the book [23,Section 11]. Below we prove Theorem 1.4.…”

Section: Proof Of Main Resultsmentioning

confidence: 99%

“…For very nice expositions of (DF )-like locally convex spaces we refer the reader to Chapter 12 of [22] and Chapter 8 of [31]. In [9], we introduced and studied the class of quasi-(DF )-spaces and constructed an example of a quasi-(DF )-space which is not a (DF )-space. The diagram below shows relationships between the discussed weak barrelledness conditions barrelled + 3…”

Section: Introductionmentioning

confidence: 99%

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Locally convex properties of free locally convex spaces

Gabriyelyan

2019

Journal of Mathematical Analysis and Applications

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Let L(X) be the free locally convex space over a Tychonoff space X. We show that the following assertions are equivalent:We prove that L(X) is a (DF )-space iff X is a countable discrete space. We show that there is a countable Tychonoff space X such that L(X) is a quasi-(DF )-space but is not a c0-quasibarrelled space. For each non-metrizable compact space K, the space L(K) is a (df )-space but is not a quasi-(DF )-space. If X is a µ-space, then L(X) has the Grothendieck property iff every compact subset of X is finite. We show that L(X) has the Dunford-Pettis property for every Tychonoff space X. If X is a sequential µ-space (for example, metrizable), then L(X) has the sequential Dunford-Pettis property iff X is discrete.2000 Mathematics Subject Classification. Primary 46A03, 46A08; Secondary 54C35.

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“…Proof. Let X be the countable Tychonoff space defined in Example 4.10 of [9]. Then, as we proved in [9], L(X) is a quasi-(DF )-space.…”

Section: Proof Of Main Resultsmentioning

confidence: 69%

Section: Proof Of Main Resultsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Locally convex properties of free locally convex spaces

Gabriyelyan

2019

Journal of Mathematical Analysis and Applications

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“…The implication (1) ⇒ (2) was recently proved by Ferrando, Gabriyelyan and Kakol [9] (with the help of cs0 * -networks). We will derive this implication from Theorem 1.2.…”

Section: Proofmentioning

confidence: 77%

$$\omega ^\omega $$-Base and infinite-dimensional compact sets in locally convex spaces

2021

Self Cite

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A locally convex space (lcs) E is said to have an $$\omega ^{\omega }$$ ω ω -base if E has a neighborhood base $$\{U_{\alpha }:\alpha \in \omega ^\omega \}$$ { U α : α ∈ ω ω } at zero such that $$U_{\beta }\subseteq U_{\alpha }$$ U β ⊆ U α for all $$\alpha \le \beta $$ α ≤ β . The class of lcs with an $$\omega ^{\omega }$$ ω ω -base is large, among others contains all (LM)-spaces (hence (LF)-spaces), strong duals of distinguished Fréchet lcs (hence spaces of distributions $$D^{\prime }(\Omega )$$ D ′ ( Ω ) ). A remarkable result of Cascales-Orihuela states that every compact set in an lcs with an $$\omega ^{\omega }$$ ω ω -base is metrizable. Our main result shows that every uncountable-dimensional lcs with an $$\omega ^{\omega }$$ ω ω -base contains an infinite-dimensional metrizable compact subset. On the other hand, the countable-dimensional vector space $$\varphi $$ φ endowed with the finest locally convex topology has an $$\omega ^\omega $$ ω ω -base but contains no infinite-dimensional compact subsets. It turns out that $$\varphi $$ φ is a unique infinite-dimensional locally convex space which is a $$k_{\mathbb {R}}$$ k R -space containing no infinite-dimensional compact subsets. Applications to spaces $$C_{p}(X)$$ C p ( X ) are provided.

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“…The implication (1) ⇒ (2) was recently proved by Ferrando, Gabriyelyan and Kakol [9] (with the help of cs * -networks). We will derive this implication from Theorem 1.2.…”

Section: Applications To Spaces C P (X)mentioning

confidence: 80%

$ω^ω$-Base and infinite-dimensional compact sets in locally convex spaces

Банах¹,

Ka̧kol²,

Phillip³

2020

Preprint

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A locally convex space (lcs) E is said to have an ω ω -base if E has a neighborhood base {U α : α ∈ ω ω } at zero such that U β ⊆ U α for all α ≤ β. The class of lcs with an ω ω -base is large, among others contains all (LM )-spaces (hence (LF )-spaces), strong duals of distinguished Fréchet lcs (hence spaces of distributions D ′ (Ω)). A remarkable result of Cascales-Orihuela states that every compact set in a lcs with an ω ω -base is metrizable. Our main result shows that every uncountable-dimensional lcs with an ω ω -base contains an infinite-dimensional metrizable compact subset. On the other hand, the countable-dimensional space ϕ endowed with the finest locally convex topology has an ω ω -base but contains no infinite-dimensional compact subsets. It turns out that ϕ is a unique infinite-dimensional locally convex space which is a k R -space containing no infinite-dimensional compact subsets. Applications to spaces C p (X) are provided.

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

Cited by 12 publications

References 40 publications

Locally convex properties of free locally convex spaces

Locally convex properties of free locally convex spaces

$$\omega ^\omega $$-Base and infinite-dimensional compact sets in locally convex spaces

$ω^ω$-Base and infinite-dimensional compact sets in locally convex spaces

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