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Journal of Convex Analysis 22 (2015), No. 4, 905--915
Copyright 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

Jerzy Kakol
Faculty of Mathematics and Informatics, A. Mickiewicz University, Matejki 48-49, 60-769 Poznan, Poland

Stephen A. Saxon
Dept. of Mathematics, University of Florida, P.O.Box 118105, Gainesville, FL 32611, U.S.A.


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

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