The topology of compact convergence on continuous function
              spaces

Warner, Seth

doi:10.1215/s0012-7094-58-02523-7

Cited by 115 publications

(12 citation statements)

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“…If C * clp * (X) is separable, then C k (X) is separable. By Theorem 5 in [21], X is submetrizable. Also, considering Theorem 4.2 in [9], it is also easy to see that X is pseudocompact since C k (X) is separable.…”

Section: Resultsmentioning

confidence: 96%

The Clp-compact-open topology on function spaces

̆glu

2017

J. of Adv. Stud. in Topol.

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Section: Resultsmentioning

confidence: 96%

The Clp-compact-open topology on function spaces

̆glu

2017

J. of Adv. Stud. in Topol.

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“…Every (DF ) (in particular, every normed) space C k (X) admits a fundamental sequence of bounded sets. By Warner [44] the space C k (X) admits a fundamental sequence of bounded sets if and only if the following condition holds:…”

Section: Proof Of Theorem 14 and Its Consequencesmentioning

confidence: 99%

On complemented copies of the space $c_0$ in spaces $C_p(X\times Y)$

Ka̧kol¹,

Marciszewski²,

Sobota³

et al. 2020

Preprint

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Cembranos and Freniche proved that for every two infinite compact Hausdorff spaces X and Y the Banach space C(X × Y ) of continuous real-valued functions on X × Y endowed with the supremum norm contains a complemented copy of the Banach space c 0 . We extend this theorem to the class of C p -spaces, that is, we prove that for all infinite Tychonoff spaces X and Y the space C p (X ×Y ) of continuous functions on X ×Y endowed with the pointwise topology contains either a complemented copy of R ω or a complemented copy of the space (c 0 ) p = {(x n ) n∈ω ∈ R ω : x n → 0}, both endowed with the product topology. We show that the latter case holds always when X × Y is pseudocompact. On the other hand, assuming the Continuum Hypothesis (or even a weaker set-theoretic assumption), we provide an example of a pseudocompact space X such that C p (X × X) does not contain a complemented copy of (c 0 ) p .As a corollary to the first result, we show that for all infinite Tychonoff spaces X and Y the space C p (X × Y ) is linearly homeomorphic to the space C p (X × Y ) × R, although, as proved earlier by Marciszewski, there exists an infinite compact space X such that C p (X) cannot be mapped onto C p (X) × R by a continuous linear surjection. This provides a positive answer to a problem of Arkhangel'ski for spaces of the form C p (X × Y ).Our another corollary-analogous to the classical Rosenthal-Lacey theorem for Banach spaces C(X) with X compact and Hausdorff-asserts that for every infinite Tychonoff spaces X and Y the space C k (X × Y ) of continuous functions on X × Y endowed with the compact-open topology admits a quotient map onto a space isomorphic to one of the following three spaces: R ω , (c 0 ) p or c 0 .

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“…Recall that locally compact spaces are completely regular and that by [Wa,Theorem 5] for a completely regular topological space Z the space C(Z) equipped with the compact-open topology is separable, if and only if Z has a separable metrizable compression, i.e. if and only if Z has a weaker separable metrizable topology.…”

Section: Continuous Functionsmentioning

confidence: 99%

Dynamics of weighted composition operators on function spaces defined by local properties

Kalmes

2019

Studia Math.

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We study topological transitivity/hypercyclicity and topological (weak) mixing for weighted composition operators on locally convex spaces of scalar-valued functions which are defined by local properties. As main application of our general approach we characterize these dynamical properties for weighted composition operators on spaces of ultradifferentiable functions, both of Beurling and Roumieu type, and on spaces of zero solutions of elliptic partial differential equations. Special attention is given to eigenspaces of the Laplace operator and the Cauchy-Riemann operator, respectively. Moreover, we show that our abstract approach unifies existing results which characterize hypercyclicity, resp. topological mixing, of (weighted) composition operators on the space of holomorphic functions on a simply connected domain in the complex plane, on the space of smooth functions on an open subset of R d , as well as results characterizing topological transitiviy of such operators on the space of real analytic functions on an open subset of R d .

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The topology of compact convergence on continuous function spaces

Cited by 115 publications

References 0 publications

The Clp-compact-open topology on function spaces

The Clp-compact-open topology on function spaces

On complemented copies of the space $c_0$ in spaces $C_p(X\times Y)$

Dynamics of weighted composition operators on function spaces defined by local properties

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