The dual of the locally convex space $C_p(X)$

J.C. Ferrando; Jerzy Kąkol; Stephen A. Saxon

doi:10.7169/facm/2014.50.2.11

June 2014 The dual of the locally convex space $C_p(X)$

J.C. Ferrando, Jerzy Kąkol, Stephen A. Saxon

Funct. Approx. Comment. Math. 50(2): 389-399 (June 2014). DOI: 10.7169/facm/2014.50.2.11

Abstract

If $X$ is an infinite Tichonov space, we show that the weak dual $L_{p}(X)$ of the continuous function space $C_{p}(X)$ cannot be barrelled, bornological, or even quasibarrelled. Indeed, of the fourteen standard weak barrelledness properties between Baire-like and primitive, $L_{p}(X)$ enjoys precisely the four between property (C) and primitive if $X$ is a P-space, and none otherwise. Since $L_{p}(X)$ is $S_{\sigma}$, it must admit an infinite-dimensional separable quotient. Under its Mackey topology, $L_{p}(X)$ enjoys eleven of the properties if $X$ is discrete, nine if $X$ is a nondiscrete P-space, and none otherwise.

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J.C. Ferrando, Jerzy Kąkol, and Stephen A. Saxon "The dual of the locally convex space $C_p(X)$," Functiones et Approximatio Commentarii Mathematici 50(2), 389-399, (June 2014). https://doi.org/10.7169/facm/2014.50.2.11

Published: June 2014

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