Weak and pointwise compactness in the space of bounded continuous functions

Wheeler, Robert F.

doi:10.1090/S0002-9947-1981-0617548-X

Weak and pointwise compactness in the space of bounded continuous functions
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Trans. Amer. Math. Soc. 266 (1981), 515-530 Request permission

Abstract:

Let $T$ be a completely regular Hausdorff space, ${C_b}(T)$ the space of bounded continuous real-valued functions on $T$, $M(T)$ the Banach space dual of ${C_b}(T)$. Let $\mathcal {H}$ denote the family of subsets of ${C_b}(T)$ which are uniformly bounded and relatively compact for the topology ${\mathfrak {J}_p}$ of pointwise convergence. The basic question considered here is: what is the largest subspace $Z$ of $M(T)$ such that every member of $\mathcal {H}$ is relatively $\sigma ({C_b},Z)$-compact? Classical results of Grothendieck and Ptak show that $Z = M(T)$ if $T$ is pseudocompact. In general, ${M_t} \subset Z \subset {M_s};$ assuming Martin’s Axiom, a deep result of Talagrand improves the lower bound to ${M_\tau }$. It is frequently, but not always, true that $Z = {M_s};$ counterexamples are given which use Banach spaces in their weak topologies to construct the underlying $T$’s.

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