
Journal of Convex Analysis 26 (2019), No. 4, 13371346 Copyright Heldermann Verlag 2019 Metrizable Bounded Sets in C(X) Spaces and Distinguished C_{p}(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, 61614 Poznan, Poland and: Institute of Mathematics, Czech Academy of Sciences, Prague, Czech Republic kakol@amu.edu.pl [Abstractpdf] Quite recently W.\,Ruess [{\it Locally convex spaces not containing $\ell_{1}$}, Funct. Approx. Comment. Math. 50 (2014) 351358] 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, FrechetUrysohn space, strong dual, strongest locally convex topology, distinguished space. MSC: 54C35, 54E15, 46A03 [ Fulltextpdf (124 KB)] for subscribers only. 