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Transactions of the American Mathematical Society

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Weak topologies on subspaces of $ C(S)$


Author: Joel H. Shapiro
Journal: Trans. Amer. Math. Soc. 157 (1971), 471-479
MSC: Primary 46E10
DOI: https://doi.org/10.1090/S0002-9947-1971-0415285-1
MathSciNet review: 0415285
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Abstract: Let $ S$ be a locally compact Hausdorff space, $ E$ a linear subspace of $ C(S)$. It is shown that the unit ball of $ E$ is compact in the strict topology if and only if both of the following two conditions are satisfied: (1) $ E$ is the Banach space dual of $ M(S)/{E^0}$ in the integration pairing, and (2) the bounded weak star topology on $ E$ coincides with the strict topology. This result is applied to several examples, among which are $ {l^\infty }$ and the space of bounded analytic functions on a plane region.


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Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1971-0415285-1
Keywords: Bounded continuous functions, bounded weak star topology, strict topology, equicontinuous set, bounded analytic functions, Lipschitz spaces
Article copyright: © Copyright 1971 American Mathematical Society

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