Journal Home Page Cumulative Index List of all Volumes Complete Contentsof this Volume Previous Article Journal of Convex Analysis 22 (2015), No. 4, 905--915Copyright Heldermann Verlag 2015 Characterizing P-spaces X in Terms of Cp(X) Juan Carlos Ferrando Centro de Investigación Operativa, Universidad Miguel Hernandez, 03202 Elche, Spain jc.ferrando@umh.es Jerzy Kakol Faculty of Mathematics and Informatics, A. Mickiewicz University, Matejki 48-49, 60-769 Poznan, Poland kakol@amu.edu.pl Stephen A. Saxon Dept. of Mathematics, University of Florida, P.O.Box 118105, Gainesville, FL 32611, U.S.A. stephen_saxon@yahoo.com [Abstract-pdf] Dual weak barrelledness led us to prove that $X$ is a $P$-space if and only if every pointwise eventually zero sequence in $C_{p}(X)$ is summable, and other better known characterizations. Novel ones recall utility functions from economics and Arkhangel'skii's (strict) $\tau$-continuity. Mackey $\aleph_0$-barrelled duality leads us to prove that $X$ is discrete if and only if every bounded $\sigma$-compact set in $C_{p}(X)$ is relatively compact. We relax the $\sigma$-compact hypothesis of Velichko and the $\sigma$-countably compact hypothesis of Tkachuk/Shakhmatov to prove\,: {\it X is a P-space if and only if $C_{p}(X)$ is $\sigma$-relatively sequentially complete}. Keywords: P-spaces, relatively compact, weak barrelledness. MSC: 54C35, 46A08 [ Fulltext-pdf  (147  KB)] for subscribers only.