ω-dominated function spaces and ω-bases in free objects of topological algebra

Банах, Тарас; Leiderman, Arkady

doi:10.1016/j.topol.2018.03.021

Cited by 22 publications

(23 citation statements)

References 30 publications

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“…The following proposition complements [, Theorem 3.2] and [, Theorem 6.5]. In fact the equivalence between (i) and (iii) below has been proved in [, Theorem 6.5] (see also [, Theorem 1.2]). The essential part is the direct proof of the implication (i) ⇒ (ii).…”

Section: More About [Fundamental] Bounded Resolutions For Spaces Cpfasupporting

confidence: 61%

“…By Ascoli's theorem , each k ‐space is Ascoli. The following proposition complements [, Theorem 3.2] and [, Theorem 6.5]. In fact the equivalence between (i) and (iii) below has been proved in [, Theorem 6.5] (see also [, Theorem 1.2]).…”

Section: More About [Fundamental] Bounded Resolutions For Spaces Cpfasupporting

confidence: 61%

“…Let us show that E satisfies (i)–(v). The space E has a

frakturG

‐base by . Clearly, Δ is a μ‐space and it has a fundamental compact sequence.…”

Section: Quasi‐(df)‐spacesmentioning

confidence: 99%

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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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Section: More About [Fundamental] Bounded Resolutions For Spaces Cpfasupporting

confidence: 61%

Section: More About [Fundamental] Bounded Resolutions For Spaces Cpfasupporting

confidence: 61%

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

Ferrando

Gabriyelyan

Ka̧kol

2019

Mathematische Nachrichten

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“…We recall that the space L(X) coincides algebraically with the space of all finitely supported sign-measures on X. Various topological and locally convex properties of free locally convex spaces are studied in [3,20,21].…”

Section: The Josefson-nissenzweig Property In Free Locally Convex Spacesmentioning

confidence: 99%

Josefson–Nissenzweig property for $$C_{p}$$Cp-spaces

Банах

Ka̧kol

Śliwa

2019

RACSAM

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We define a locally convex space E to have the Josefson-Nissenzweig property (JNP) if the identity map (E ′ , σ(E ′ , E)) → (E ′ , β * (E ′ , E)) is not sequentially continuous. By the classical Josefson-Nissenzweig theorem, every infinite-dimensional Banach space has the JNP.We show that for a Tychonoff space X, the function space Cp(X) has the JNP iff there is a weak * null-sequence {µn}n∈ω of finitely supported sign-measures on X with unit norm. However, for every Tychonoff space X, neither the space B1(X) of Baire-1 functions on X nor the free locally convex space L(X) over X has the JNP. We also define two modifications of the JNP, called the universal JNP and the JNP everywhere (briefly, the uJNP and eJNP), and thoroughly study them in the classes of locally convex spaces, Banach spaces and function spaces. We provide a characterization of the JNP in terms of operators into locally convex spaces with the uJNP or eJNP and give numerous examples clarifying relationships between the considered notions.2010 Mathematics Subject Classification. Primary 46A03; Secondary 46E10, 46E15.

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“…A description of the topology µ µ µ X of V(X) for a uniform space X is given in Section 5 of [3]. In the next theorem we describe the topologies µ µ µ X and ν ν ν X for any Tychonoff space X.…”

Section: Theorem 21 ([33]mentioning

confidence: 99%

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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ω-dominated function spaces and ω-bases in free objects of topological algebra

Cited by 22 publications

References 30 publications

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

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

Josefson–Nissenzweig property for $$C_{p}$$Cp-spaces

Locally convex properties of free locally convex spaces

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