Analogs of the Krein-Milman theorem for bounded convex sets in infinite-dimensional spaces (Q2791916)

Let \(U\) be a closed absolutely convex absorbing subset of a Banach space \(E\). By \(p_U\) denote the Minkowski functional of \(U\). Assume that \(p_U\) is a norm on \(E\) (or, equivalently, that \(U\) does not contain a non-trivial linear subspace). Denote by \(E_U\) the completion of \((E, p_U)\) and denote by \(J_U\) the natural embedding of \(E\) into \(E_U\).NEWLINENEWLINEAccording to the author's definition, such a subset \(U\) is said to be anti-compact if for every bounded subset \(A \subset E\) its image \(J_U(A)\) is pre-compact in \(E_U\). The author proves that a Banach space \(E\) possesses an anti-compact subset if and only if the dual space \(E^*\) contains a countable total subset (or, in other words, iff \(E^*\) is \(w^*\)-separable).NEWLINENEWLINEDenote by \(\bar C(E)\) the collection of all anti-compact subsets of \(E\). The main result of the paper says that if \(\bar C(E) \neq \emptyset\), then for every nonempty bounded convex closed subset \(A \subset E\) the following formula holds true: NEWLINE\[NEWLINE A = \bigcap_{U \in \bar C(E)} \overline{J_U^{-1}\;(J_U(E) \cap \overline{\mathrm{conv}} \;\mathrm{ext} \overline{J_U(A))})}, NEWLINE\]NEWLINE where \(\overline{\mathrm{conv}}\) and \(\mathrm{ext}\) stand for the closed convex hull and the set of extreme points, respectively.