An extension of the Krein–Smulian theorem. (English) Zbl 1117.46002

Granero, Antonio S.

Rev. Mat. Iberoam. 22, No. 1, 93-110 (2006).

It is proved that if \(Z\) is a closed subspace of a Banach space \(X\) and \(K\) is a weak* compact subset of the bidual \(X^{**}\), then the distance from the weak* closed convex hull of \(K\) to \(Z\) is at most 5 times the distance from \(K\) to \(Z\). In the special case \(K\subset Z=X\), we recover the classical Krein–Šmulian theorem. In many cases, these two distances are equal. This is proved in particular for weakly compactly generated Banach spaces, and for spaces whose duals do not contain \(\ell_1\). An example is given where the two distances are different, but it depends on the continuum hypothesis.
Since this paper was written, the author, P.Hájek and V.Montesinos Santalucía [Math.Ann.328, No.4, 625–631 (2004; Zbl 1059.46015)] have published examples which do not require the continuum hypothesis.

Reviewer: David Yost (Ballarat)

MSC:

46A50	Compactness in topological linear spaces; angelic spaces, etc.
46B26	Nonseparable Banach spaces

Keywords:

closed convex hull; Krein-Šmulian theorem

Citations:

Zbl 1059.46015

References:

[1]	Argyros, S., Mercourakis, S. and Negrepontis, S.: Functional- analytic properties of Corson-compact spaces. Studia Math. 89 (1988), 197-229. · Zbl 0656.46014
[2]	Argyros, S. and Mercourakis, S.: On weakly Lindelöf Banach spaces. Rocky Mountain J. Math. 23 (1993), 395-446. · Zbl 0797.46009
[3]	Choquet, G.: Lectures on analysis. Vol. II: Representation theory. A. Benjamin, Inc., New York-Amsterdam, 1969. · Zbl 0181.39602
[4]	Comfort, W. W. and Negrepontis, S. A.: Chain Conditions in topol- ogy. Cambridge Tracts in Math. 79. Cambridge Univ. Press, 1982. · Zbl 0488.54002
[5]	Diestel, J. and Uhl, J. J.: Vector measures. Mathematical Surveys 15. American Mathematical Society, Providence, R.I., 1977. · Zbl 0369.46039
[6]	Fabian, M.: G\hat ateaux differentiability of convex functions and topology. Weak Asplund Spaces. Canadian Mathematical Society Series of Mono- graphs and Advanced Texts. A Wiley-Interscience Publication. John Wiley & Sons, Inc., New York, 1997. · Zbl 0883.46011
[7]	Fabian, M., Hájek, P., Montesinos, V. and Zizler, V.: A quanti- tative version of Krein’s Theorem. Rev. Mat. Iberoamericana 21 (2005), no. 1, 237-248. · Zbl 1083.46012
[8]	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