Characterizing P-spaces X in Terms of C_p(X)

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 : X is a P-space if and only if $C_{p}(X)$ is $\sigma$-relatively sequentially complete.