Tightness and distinguished Fréchet spaces

Ferrando, J. C.; Ka̧kol, Jerzy; Pellicer, Manuel López; Saxon, Stephen A.

doi:10.1016/j.jmaa.2005.12.059

Cited by 31 publications

(31 citation statements)

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“…Following [15], we will say that a base {U p : p ∈ P} of neighborhoods of zero of a locally convex space E is a G-base if U q ⊂ U p whenever p ≤ q.…”

Section: Resolutions and Completenessmentioning

confidence: 99%

Compact covers and function spaces

Ka̧kol

Pellicer

Okunev

2014

Journal of Mathematical Analysis and Applications

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Abstract. For a Tychonoff space X, we denote by C p (X) and C c (X) the space of continuous real-valued functions on X equipped with the topology of pointwise convergence and the compact-open topology respectively.Providing a characterization of the Lindelöf Σ-property of X in terms of C p (X), we extend Okunev's results by showing that if there exists a surjection from) that takes bounded sequences to bounded sequences, then υY is a Lindelöf Σ-space (respectively K-analytic) if υX has this property. In the second part, applying Christensen's theorem, we extend Pelant's result by proving that if X is a separable completely metrizable space and Y is first countable, and there is a quotient linear map from C c (X) onto C c (Y ), then Y is a separable completely metrizable space. We study also the non-separable case, and consider a different approach to the result of J. Baars, J. de Groot, J. Pelant and V. Valov, which is based on the combination of two facts: Complete metrizability is preserved by p -equivalence in the class of metric spaces (J. Baars, J. de Groot, J. Pelant). If X is completely metrizable and p -equivalent to a first countable Y , then Y is metrizable (V. Valov). Some additional results are presented.

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“…Following [15], we will say that a base {U p : p ∈ P} of neighborhoods of zero of a locally convex space E is a G-base if U q ⊂ U p whenever p ≤ q.…”

Section: Resolutions and Completenessmentioning

confidence: 99%

Compact covers and function spaces

Ka̧kol

Pellicer

Okunev

2014

Journal of Mathematical Analysis and Applications

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“…Last Proposition 13 has been used [24] to study non distinguished Köthe echelon spaces. We showed in [35] also the following…”

Section: Proposition 13 a Fréchet Space E Is Distinguished If And Onmentioning

confidence: 99%

On realcompact topological vector spaces

Ka̧kol

Pellicer

2011

RACSAM

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This survey paper collects some of older and quite new concepts and results from descriptive set topology applied to study certain infinite-dimensional topological vector spaces appearing in Functional Analysis, including Fréchet spaces, (L F)-spaces, and their duals, (D F)-spaces and spaces of continuous real-valued functions C(X ) on a completely regular Hausdorff space X . Especially (L F)-spaces and their duals arise in many fields of Functional Analysis and its applications, for example in Distributions Theory, DifferentialEquations and Complex Analysis. The concept of a realcompact topological space, although originally introduced and studied in General Topology, has been also studied because of very concrete applications in Linear Functional Analysis.

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“…But E a := (E, a(E, E')) does not belong to class 0. Otherwise, since E a is dense in the product R 1 for some set /, the space R 7 would also belong to class (9,[6,Proposition 8]. But R 7 is a Baire space, so [7,Theorem 4.8] applies to show that R 7 has countable tightness, that is, for every AcW and every x € A there exists a countable set B C A such that x € B. Consequently, / is countable and E is finite-dimensional, a contradiction.…”

Section: Corollary 5 the Strong Dual (E' $(E\ E)) Of A Locally Conmentioning

confidence: 99%

“…The class <S contains (among the others) (LAf)-spaces (hence (LF)-spaces), the dual metric spaces (hence (DF)-spaces), the space of distributions D'(Cl) and the space A(Q.) of the real analytic functions for open fi C R N (see [8,9]). By C C (X) we denote the space of all real-valued continuous functions on a completely regular space X equipped with the compact-open topology.…”

Section: Introductionmentioning

confidence: 99%