Functiones et Approximatio Commentarii Mathematici

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

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

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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.

Article information

Funct. Approx. Comment. Math. Volume 50, Number 2 (2014), 389-399.

First available in Project Euclid: 26 June 2014

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 46A08: Barrelled spaces, bornological spaces
Secondary: 54C35: Function spaces [See also 46Exx, 58D15]

weak barrelledness P-spaces separable quotients


Ferrando, J.C.; Kąkol, Jerzy; Saxon, Stephen A. The dual of the locally convex space $C_p(X)$. Funct. Approx. Comment. Math. 50 (2014), no. 2, 389--399. doi:10.7169/facm/2014.50.2.11.

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