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The Krein-┼ámulian theorem and its extensions. (English) Zbl 1192.46015

This survey paper describes some topics in the author’s work on quantitative versions of the Krein-Shmulyan theorem, often with full proofs.
The theorem in question is the following. If \(K\) is a weakly compact subset of a Banach space \(X\), then the closed convex hull of \(K\) is weakly compact, too. This leads to the conjecture that for a weak\(^*\) compact subset \(K\) of \(X^{**}\), the distance \(d\) from \(K\) to \(X\), i.e., \(d= \sup\{ \inf \{\|k-x\|: x\in X\} : k\in K\}\), coincides with the distance \(d^*\) of the weak\(^*\)-closed convex hull of \(K\) to \(X\).
Here is a small sample of the many results in this paper. In general \(d<d^*\), so the conjecture is false; but for certain classes of Banach spaces one has in fact \(d=d^*\). This is particularly true for separable spaces, although this information is a little hidden (cf. Proposition 4.9). On the other hand the author establishes the inequality \(d^*\leq 5d\) for all \(X\), and he gives examples where \(d^*\geq 3d>0\).

MSC:

46B26 Nonseparable Banach spaces
46A55 Convex sets in topological linear spaces; Choquet theory
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