A Banach-Dieudonné theorem for the space of bounded continuous functions on a separable metric space with the strict topology

Richard C. Kraaij

doi:10.1016/j.topol.2016.06.003

A Banach-Dieudonné theorem for the space of bounded continuous functions on a separable metric space with the strict topology

Richard C. Kraaij^*

^*Corresponding author for this work

Applied Probability

Research output: Contribution to journal › Article › Scientific › peer-review

3 Citations (Scopus)

105 Downloads (Pure)

Abstract

Let X be a separable metric space and let β be the strict topology on the space of bounded continuous functions on X, which has the space of τ-additive Borel measures as a continuous dual space. We prove a Banach-Dieudonné type result for the space of bounded continuous functions equipped with β: the finest locally convex topology on the dual space that coincides with the weak topology on all weakly compact sets is a k-space. As a consequence, the space of bounded continuous functions with the strict topology is hypercomplete and a Pták space. Additionally, the closed graph, inverse mapping and open mapping theorems holds for linear maps between space of this type.

Original language	English
Pages (from-to)	181-188
Number of pages	8
Journal	Topology and Its Applications: a journal devoted to general, geometric, set-theoretic and algebraic topology
Volume	209
DOIs	https://doi.org/10.1016/j.topol.2016.06.003
Publication status	Published - 15 Aug 2016

Keywords

Banach-Dieudonné theorem
Closed graph theorem
Space of bounded continuous functions
Strict topology

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10.1016/j.topol.2016.06.003

7548513Accepted author manuscript, 272 KB

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title = "A Banach-Dieudonn{\'e} theorem for the space of bounded continuous functions on a separable metric space with the strict topology",

abstract = "Let X be a separable metric space and let β be the strict topology on the space of bounded continuous functions on X, which has the space of τ-additive Borel measures as a continuous dual space. We prove a Banach-Dieudonn{\'e} type result for the space of bounded continuous functions equipped with β: the finest locally convex topology on the dual space that coincides with the weak topology on all weakly compact sets is a k-space. As a consequence, the space of bounded continuous functions with the strict topology is hypercomplete and a Pt{\'a}k space. Additionally, the closed graph, inverse mapping and open mapping theorems holds for linear maps between space of this type.",

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A Banach-Dieudonné theorem for the space of bounded continuous functions on a separable metric space with the strict topology. / Kraaij, Richard C.
In: Topology and Its Applications: a journal devoted to general, geometric, set-theoretic and algebraic topology, Vol. 209, 15.08.2016, p. 181-188.

Research output: Contribution to journal › Article › Scientific › peer-review

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T1 - A Banach-Dieudonné theorem for the space of bounded continuous functions on a separable metric space with the strict topology

AU - Kraaij, Richard C.

PY - 2016/8/15

Y1 - 2016/8/15

N2 - Let X be a separable metric space and let β be the strict topology on the space of bounded continuous functions on X, which has the space of τ-additive Borel measures as a continuous dual space. We prove a Banach-Dieudonné type result for the space of bounded continuous functions equipped with β: the finest locally convex topology on the dual space that coincides with the weak topology on all weakly compact sets is a k-space. As a consequence, the space of bounded continuous functions with the strict topology is hypercomplete and a Pták space. Additionally, the closed graph, inverse mapping and open mapping theorems holds for linear maps between space of this type.

AB - Let X be a separable metric space and let β be the strict topology on the space of bounded continuous functions on X, which has the space of τ-additive Borel measures as a continuous dual space. We prove a Banach-Dieudonné type result for the space of bounded continuous functions equipped with β: the finest locally convex topology on the dual space that coincides with the weak topology on all weakly compact sets is a k-space. As a consequence, the space of bounded continuous functions with the strict topology is hypercomplete and a Pták space. Additionally, the closed graph, inverse mapping and open mapping theorems holds for linear maps between space of this type.

KW - Banach-Dieudonné theorem

KW - Closed graph theorem

KW - Space of bounded continuous functions

KW - Strict topology

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JO - Topology and Its Applications: a journal devoted to general, geometric, set-theoretic and algebraic topology

JF - Topology and Its Applications: a journal devoted to general, geometric, set-theoretic and algebraic topology

ER -

A Banach-Dieudonné theorem for the space of bounded continuous functions on a separable metric space with the strict topology

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Corrigendum to: ‘A Banach–Dieudonné theorem for the space of bounded continuous functions on a separable metric space with the strict topology’ (Topology and its Applications (2016) 209 (181–188), (S0166864116301213) (10.1016/j.topol.2016.06.003))

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