On nested sequences of convex sets in Banach spaces
Let \(X\) be a real or complex Banach space and \((C_n)\) be a nested sequence of nonempty closed, bounded convex subsets of \(X\). By a surprisingly elementary argument it is established that (Theorem 1) \[ \forall x\in X,\;\lim_{n\to\infty} d(x,C_n)=d\left(x,\bigcap_{n}\overline{C_n}^{w^\ast}\right). \] This result was previously proved for separable spaces by \textit{J. M. F. Castillo} and \textit{P. L. Papini} [``Approximation of the limit distance function in Banach spaces'', J. Math. Anal. Appl. 328, No. 1, 577--589 (2007; Zbl 1119.46017)], where it is also shown that convexity is essential. If \((C_n)\) is contained in some weakly compact set \(C\) of \(X\), then of course the above result has to be true without any \(w^\ast\)-closure. Indeed (Theorem 2), \(C\) is weakly compact if and only if the above theorem holds without \(w^\ast\)-closure for all choices of \((C_n)\) inside \(C\). Here James's compactness theorem plays a key role for the proof. The remaining part concerns two related questions on approximating \(w^*\)-compact, convex sets in \(X^{\ast\ast}\) when \(X^\ast\) is separable, posed by Castillo and Papini. The author obtains (Theorem 3) that when \(X^\ast\) is separable, then for every \(w^*\)-compact, convex set \(C\) in \(X^{\ast\ast}\), there exists a nested sequence \((C_n)\) of closed, bounded and convex sets in \(X\) such that \(C=\bigcap_n \overline{C_n}^{w^\ast}\).
scientific article; zbMATH DE number 6009681