Free Locally Convex Spaces and the<i>k</i>-space Property

Gabriyelyan, Saak

doi:10.4153/cmb-2014-019-7

Cited by 13 publications

(7 citation statements)

References 14 publications

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“…In [12] the first named author proved that the free lcs L(X) over a Tychonoff space X is a k-space if and only if X is a discrete countable space. Question 6.11.…”

Section: Proof Of Theorem 110 and Final Questionsmentioning

confidence: 99%

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The Ascoli property for function spaces and the weak topology of Banach and Fréchet spaces

2016

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Following [3] we say that a Tychonoff space X is an Ascoli space if every compact subset K of C k (X) is evenly continuous; this notion is closely related to the classical Ascoli theorem. Every k R -space, hence any k-space, is Ascoli.Let X be a metrizable space. We prove that the space C k (X) is Ascoli iff C k (X) is a k R -space iff X is locally compact. Moreover, C k (X) endowed with the weak topology is Ascoli iff X is countable and discrete.Using some basic concepts from probability theory and measure-theoretic properties of ℓ 1 , we show that the following assertions are equivalent for a Banach space E: (i) E does not contain isomorphic copy of ℓ 1 , (ii) every real-valued sequentially continuous map on the unit ball B w with the weak topology is continuous,We prove also that a Fréchet lcs F does not contain isomorphic copy of ℓ 1 iff each closed and convex bounded subset of F is Ascoli in the weak topology. However we show that a Banach space E in the weak topology is Ascoli iff E is finite-dimensional. We supplement the last result by showing that a Fréchet lcs F which is a quojection is Ascoli in the weak topology iff either F is finite dimensional or F is isomorphic to the product K N , where K ∈ {R, C}.

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“…In [12] the first named author proved that the free lcs L(X) over a Tychonoff space X is a k-space if and only if X is a discrete countable space. Question 6.11.…”

Section: Proof Of Theorem 110 and Final Questionsmentioning

confidence: 99%

“…We do not know the answer even if "Ascoli" is replaced by a stronger assumption "L(X) is a k R -space" (see [12,Question 3.6]).…”

Section: Proof Of Theorem 110 and Final Questionsmentioning

confidence: 99%

The Ascoli property for function spaces and the weak topology of Banach and Fréchet spaces

2016

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“…Then is compact by Proposition 7, and so is closed. The converse assertion follows from Lemma 3.3 of [16].…”

Section: Theorem 8 Let Be a Subspace Of A Compact Metrizable Space mentioning

confidence: 63%

Free Subspaces of Free Locally Convex Spaces

Gabriyelyan

Morris

2018

Journal of Function Spaces

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If and are Tychonoff spaces, let ( ) and ( ) be the free locally convex space over and , respectively. For general and , the question of whether ( ) can be embedded as a topological vector subspace of ( ) is difficult. The best results in the literature are that if ( ) can be embedded as a topological vector subspace of (I), where I = [0, 1], then is a countabledimensional compact metrizable space. Further, if is a finite-dimensional compact metrizable space, then ( ) can be embedded as a topological vector subspace of (I). In this paper, it is proved that ( ) can be embedded in (R) as a topological vector subspace if is a disjoint union of a countable number of finite-dimensional locally compact separable metrizable spaces. This is the case if = R , ∈ N. It is also shown that if G and denote the Cantor space and the Hilbert cube I N , respectively, then (i) ( ) is embedded in (G) if and only if is a zero-dimensional metrizable compact space; (ii) ( ) is embedded in ( ) if and only if is a metrizable compact space.

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“…Especially important properties are those ones which generalize metrizability: the Fréchet-Urysohn property, sequentiality, the k-space property and countable tightness. These properties are intensively studied for function spaces, free locally convex spaces, (LM )-spaces, strict (LF )-spaces and their strong duals, and Banach and Fréchet spaces in the weak topology etc., see for example [1,10,23,24,26,31] and references therein.…”

Section: Introductionmentioning

confidence: 99%

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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Free Locally Convex Spaces and thek-space Property

Abstract: Abstract. Let L(X) be the free locally convex space over a Tychonoff space X. Then L(X) is a k-space if and only if X is a countable discrete space. We prove also that L(D) has uncountable tightness for every uncountable discrete space D.

Cited by 13 publications

References 14 publications

The Ascoli property for function spaces and the weak topology of Banach and Fréchet spaces

The Ascoli property for function spaces and the weak topology of Banach and Fréchet spaces

Free Subspaces of Free Locally Convex Spaces

Ascoli and sequentially Ascoli spaces

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