Abstract
We saw in the previous chapter that regardless of the normed linear space X, weak* closed, bounded sets in X* are weak* compact. How does a subset K of a Banach space X get to be weakly compact? The two are related. Before investigating their relationship, we look at a couple of necessary ingredients for weak compactness and take a close look at two illustrative nonweakly compact sets.
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Bibliography
Bourgin, D. G. 1942. Some properties of Banach spaces. Amer. J. Math., 64, 597 – 612.
Brace, J. W. 1955. Compactness in the weak topology. Math. Mag., 28, 125 – 134.
Eberlein, W. F. 1947. Weak compactness in Banach spaces, I. Proc. Nat. Acad. Sci. USA, 33, 51 – 53.
Grothendieck, A. 1952. Critfères de compacité dans les espaces fonctionnels généraux. Amer. J. Math., 74, 168 – 186.
Floret, K. 1980. Weakly Compact Sets. Springer Lecture Notes in Mathematics, Volume 801. New York: Springer-Verlag.
Lindenstrauss, J. 1972. Weakly compact sets—their topological properties and the Banach spaces they generate. Proceedings of the Symposium on Infinite Dimensional Topology, Annals of Math. Studies, no. 69, 235 – 263.
Pelczynski, A. 1964. A proof of Eberlein-Smulian theorem by an application of basic sequences. Bull. Acad. Polon. Sci., 12, 543 – 548.
Phillips, R. S. 1943. On weakly compact subsets of a Banach space. Amer. J. Math., 65, 108 – 136.
Smulian, V. L. 1940. Über lineare topologische Räume. Mat. Sbornik N.S., 7(49), 425 – 448.
Whitley, R. J. 1967. An elementary proof of the Eberlein-Smulian theorem. Math. Ann., 172, 116 – 118.
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© 1984 Springer-Verlag New York, Inc.
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Diestel, J. (1984). The Eberlein-Šmulian Theorem. In: Sequences and Series in Banach Spaces. Graduate Texts in Mathematics, vol 92. Springer, New York, NY. https://doi.org/10.1007/978-1-4612-5200-9_3
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DOI: https://doi.org/10.1007/978-1-4612-5200-9_3
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