## The Krein-Šmulian 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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