Journal of Convex Analysis 26 (2019), No. 4, 1337--1346
Copyright Heldermann Verlag 2019
Metrizable Bounded Sets in C(X) Spaces and Distinguished Cp(X) Spaces
Juan Carlos Ferrando
Centro de Investigación Operativa, Universidad Miguel Hernández, 03202 Elche, Spain
jc.ferrando@umh.es
Jerzy Kakol
Faculty of Mathematics and Informatics, A. Mickiewicz University, 61-614 Poznan, Poland
and: Institute of Mathematics, Czech Academy of Sciences, Prague, Czech Republic
kakol@amu.edu.pl
[Abstract-pdf]
Quite recently W.\,Ruess [{\it Locally convex spaces not containing $\ell_{1}$}, Funct. Approx. Comment. Math. 50 (2014) 351--358] has shown that a wide class of locally convex spaces for which all bounded sets are metrizable enjoy Rosenthal's $\ell_{1} $-dichotomy. Being motivated by this fact we show that for a Tychonoff space $X$ the bounded sets of $C_{p}(X) $ are metrizable (respectively, the bounded sets of $C_{k}(X)$ are weakly metrizable) if and only if $X$ is countable. If $X$ is a $P$-space we show that every bounded set in $C_{p}(X) $ is metrizable if and only if $X$ is countable and discrete. The second part of the paper deals with distinguished $C_{p}(X) $ spaces. Among other things we show that $C_{p}(X) $ is distinguished if and only if the strong topology of the dual coincides with its strongest locally convex topology, and that $C_{p}(X)$ is always distinguished whenever $X$ is countable.
Keywords: Countable tightness, Frechet-Urysohn space, strong dual, strongest locally convex topology, distinguished space.
MSC: 54C35, 54E15, 46A03
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