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Weak Compactness and Separation

Published online by Cambridge University Press:  20 November 2018

Robert C. James*
Affiliation:
Harvey Mudd College and Institute for Advanced Study, Princeton, New Jersey
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The purpose of this paper is to develop characterizations of weakly compact subsets of a Banach space in terms of separation properties. The sets A and B are said to be separated by a hyperplane H if A is contained in one of the two closed half-spaces determined by H, and B is contained in the other; A and B are strictly separated by H if A is contained in one of the two open half-spaces determined by H, and B is contained in the other. The following are known to be true for locally convex topological linear spaces.

Type
Research Article
Copyright
Copyright © Canadian Mathematical Society 1964

References

1. Bourbaki, N., Espaces vectoriels topologiques, Actualités Sci. Ind., No. 489 (Paris, 1953).Google Scholar
2. Day, M. M., Normed linear spaces (Berlin, 1958).Google Scholar
3. James, R. C., Weakly compact sets, submitted for publication in Trans. Amer. Math. Soc.Google Scholar
4. Klee, V. L. Jr., Convex sets in linear spaces, Duke Math. J., 18 (1951), 443466, 875-883.Google Scholar
5. Tukey, J. W., Some notes on the separation of convex sets, Portugaliae Math., 8 (1942), 95 102.Google Scholar