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The authors note that for separable spaces this result was proved by H. P. Rosenthal in an unpublished paper.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)].


46B50 Compactness in Banach (or normed) spaces
46A50 Compactness in topological linear spaces; angelic spaces, etc.
Full Text: DOI EuDML


[1] Bellenot, S. F., Haydon, R. and Odell, E.: Quasi-reflexive and tree spaces constructed in the spirit of R. C. James. In Banach space theory (Iowa City, IA, 1987), 19-43. Contemp. Math. 85. Amer. Math. Soc., Prov- idence, RI, 1989. · Zbl 0726.46005
[2] Deville, R., Godefroy, G. and Zizler, V.: Smoothness and Renorm- ings in Banach Spaces. Pitman Monographs and Surveys in Pure and Ap- plied Mathematics 64. Longman Scientific & Technical, Harlow, 1993. · Zbl 0782.46019
[3] Diestel, J.: Sequences and Series in Banach Spaces. Graduate Texts in Mathematics 92. Springer-Verlag, New York, 1984. · Zbl 0542.46007
[4] Eberlein, W. F.: Weak compactness in Banach spaces, I. Proc. Natl. Acad. Sci. USA 33 (1947), 51-53. · Zbl 0029.26902
[5] Fabian, M., Montesinos, V. and Zizler, V.: A characterization of subspaces of weakly compactly generated Banach spaces. J. London Math. Soc. (2) 69 (2004), no. 2, 457-464. · Zbl 1059.46014
[6] Fabian, M., Habala, P., Hájek, P., Pelant, J., Montesinos, V. and Zizler, V.: Functional Analysis and Infinite Dimensional Topology. CMS Books in Mathematics 8. Springer Verlag, New York, 2001. · Zbl 0981.46001
[7] Godefroy, G. and Kalton, N.; The ball topology and its applications. In Banach space theory (Iowa City, IA, 1987), 195-237. Contemp. Math. 85. Amer. Math. Soc., Providence, RI, 1989. · Zbl 0676.46003
[8] Granero, A. S.: An extension of the Krein-\check Smulyan Theorem in Banach spaces. To appear in Rev. Mat. Iberoamericana.
[9] Granero, A. S., Háyek, P. and Montesinos, V.: Convexity and w\ast -compactness in Banach spaces. Math. Ann. 328 (2004), 625-631. · Zbl 1059.46015
[10] Grothendieck, A.: Crit‘eres de compacité dans les espaces fonctionnels généraux. Amer. J. Math. 74 (1952), 168-186. · Zbl 0046.11702
[11] Köthe, G.: Topological Vector Spaces I. Die Grundlehren der mathema- tischen Wissenschaften 159. Springer-Verlag, New York, 1969.
[12] Pták, V.: A combinatorial lemma on the existence of convex means and its applications to weak compactness. In 1963 Proc. Sympos. Pure Math. vol. VIII, 437-450. Amer. Math. Soc. Providence, RI, 1963. · Zbl 0144.16903
[13] Rosenthal, H. P.: The heredity problem for weakly compactly generated Banach spaces. Compositio Math. 28 (1974), 83-111. · Zbl 0298.46013
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