Characterizing P-spaces X in Terms of C _p (X)

Dual weak barrelledness led us to prove that

is a

P P

-space if and only if every pointwise eventually zero sequence in

C_{p}(X) C p ( X )

is summable, and other better known characterizations. Novel ones recall utility functions from economics and Arkhangel'skii's (strict)

\tau τ

-continuity. Mackey

\aleph_0 ℵ 0

-barrelled duality leads us to prove that

X X

is discrete if and only if every bounded

\sigma σ

-compact set in

C_{p}(X) C p ( X )

is relatively compact. We relax the

\sigma σ

-compact hypothesis of Velichko and the

\sigma σ

-countably compact hypothesis of Tkachuk/Shakhmatov to prove : X is a P-space if and only if

C_{p}(X) C p ( X )

is

\sigma σ

-relatively sequentially complete.