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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

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


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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