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A quantitative version of Krein’s theorem. (English) Zbl 1083.46012

Motivated by the classical Kreĭn theorem on weak compactness of the closed convex hull of a weakly compact set, the authors say that a bounded subset \(M\) of a Banach space \(X\) is \(\varepsilon\)-weakly relatively compact (\(\varepsilon\)-WRC) if \(\overline{M}^{w^*}\subset X+\varepsilon B_{X^{**}}\). They prove that for every \(\varepsilon\)-WRC set \(M\), its convex hull is \(2\varepsilon\)-WRC. In a large class of Banach spaces, e.g., if \(B_{X^*}\) is weakly* angelic, we can put in this statement \(\varepsilon\)-WRC instead of \(2\varepsilon\)-WRC.
The authors note that for separable spaces this result was proved by H. P. Rosenthal in an unpublished paper. Some applications to weakly compactly generated Banach spaces are given. See also A. S. Granero [Rev. Mat. Iberoam. 22, No. 1, 93–110 (2006; Zbl 1117.46002)] and A. S. Granero, P. Hájek and V. Montesinos Santalucia [Math. Ann. 328, 625–631 (2004; Zbl 1059.46015)].

MSC:

46B50 Compactness in Banach (or normed) spaces
46A50 Compactness in topological linear spaces; angelic spaces, etc.
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References:

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