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Weak and pointwise compactness in the space of bounded continuous functions


Author: Robert F. Wheeler
Journal: Trans. Amer. Math. Soc. 266 (1981), 515-530
MSC: Primary 46E15; Secondary 46E27
DOI: https://doi.org/10.1090/S0002-9947-1981-0617548-X
MathSciNet review: 617548
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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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DOI: https://doi.org/10.1090/S0002-9947-1981-0617548-X
Article copyright: © Copyright 1981 American Mathematical Society

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