Locally convex properties of free locally convex spaces

Gabriyelyan, Saak

doi:10.1016/j.jmaa.2019.123453

Cited by 20 publications

(60 citation statements)

References 101 publications

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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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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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“…was intensively studied in [3,5,11,13,14,16,17,18,19]. Being motivated by the classical notion of c 0 -barrelled locally convex spaces, in [15] we defined a Tychonoff space X to be sequentially Ascoli if every convergent sequence in C k (X) is equicontinuous. Clearly, every Ascoli space is sequentially Ascoli, but the converse is not true in general, see [15].…”

Section: Introductionmentioning

confidence: 99%

“…Being motivated by the classical notion of c 0 -barrelled locally convex spaces, in [15] we defined a Tychonoff space X to be sequentially Ascoli if every convergent sequence in C k (X) is equicontinuous. Clearly, every Ascoli space is sequentially Ascoli, but the converse is not true in general, see [15]. Below we formulate some of the most interesting results (although the clauses (i)-(iv) were proved for the property of being an Ascoli space, their proofs and Proposition 2.10 show that one can replace "Ascoli" by "sequentially Ascoli").…”

Section: Introductionmentioning

confidence: 99%

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Ascoli and sequentially Ascoli spaces

Gabriyelyan¹

2020

Preprint

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A Tychonoff space X is called (sequentially) Ascoli if every compact subset (resp. convergent sequence) of C k (X) is evenly continuous, where C k (X) denotes the space of all real-valued continuous functions on X endowed with the compact-open topology. Various properties of (sequentially) Ascoli spaces are studied, and we give several characterizations of sequentially Ascoli spaces. Strengthening a result of Arhangel'skii we show that a hereditary Ascoli space is Fréchet-Urysohn. A locally compact abelian group G with the Bohr topology is sequentially Ascoli iff G is compact. If X is totally countably compact or near sequentially compact then it is a sequentially Ascoli space. The product of a locally compact space and an Ascoli space is Ascoli. If additionally X is a µ-space, then X is locally compact iff the product of X with any Ascoli space is an Ascoli space. Extending one of the main results of [18] and [16] we show that C p (X) is sequentially Ascoli iff X has the property (κ). We give a necessary condition on X for which the space C k (X) is sequentially Ascoli. For every metrizable abelian group Y , Y -Tychonoff space X, and nonzero countable ordinal α, the space B α (X, Y ) of Baire-α functions from X to Y is κ-Fréchet-Urysohn and hence Ascoli.

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“…By [4, 5.4] (and [4, 5.8]), a (Tychonoff) space X is Ascoli if and only if the canonical map δ : X → C k (C k (X)) assigning to each x ∈ X the Dirac measure δ x : f → f (x) is continuous (if and only if the map δ is a topological embedding). Sequentially Ascoli spaces were studied in [14], [15] and in [3] (as spaces containing no strict Cld ω -fans). For any Tychonoff space X we have the implications:…”

Section: Introductionmentioning

confidence: 99%

The cometrizability of generalized metric spaces

Банах¹,

Stelmakh²

2019

Preprint

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A topological space X is cometrizable if it admits a weaker metrizable topology such that each point x ∈ X has a (not necessarily open) neighborhood base consisting of metrically closed sets. We prove that the class of cometrizable spaces includes all stratifiable spaces and all as-cosmic spaces. Moreover, for any k-separable space X and any cometrizable space Y , the function space C k (X, Y ) endowed with the compact-open topology is cometrizable. For a topological space X the function space Cp(X) is cometrizable if and only if it is metrizable. Also, we present an example of a regular countable space of weight ω1, which is not cometrizable. Under ω1 = c this space contains no infinite compact subsets and hence is cs-cosmic. Under ω1 < p this countable space is Fréchet-Urysohn and is not cs-cosmic.

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Locally convex properties of free locally convex spaces

Cited by 20 publications

References 101 publications

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

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

Ascoli and sequentially Ascoli spaces

The cometrizability of generalized metric spaces

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