Weak topologies on subspaces of C(S)


Joel H. Shapiro


Trans. Amer. Math. Soc. 157 (1971), 471--479.
Abstract: This paper concerns subspaces E of C(S), where S is locally compact and Hausdorff. The main result is that the unit ball of E is compact in the strict topology iff E is the Banach space dual (in the integration pairing) of the quotient space M(S)/E^0, and the bounded weak star topolgy on E coincides with the strict topology. This result is applied to several examples, among which are the space of bounded sequences and the space of bounded analytic functions on a plane region.
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