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

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.


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


Zbl 1059.46015
Full Text: DOI EuDML


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