A Study on the \(\mathscr{F}\)-Simultaneous Approximative \(\tau\)-Compactness Property in Banach Spaces

Abstract

Veselý (1997) studied Banach spaces that admit \(f\)-centers for finite subsets of the space. In this work, we introduce the concept of the \(\mathscr{F}\)-simultaneous approximative \(\tau\)-compactness property (\(\tau\)-\(\mathscr{F}\)-\(\mathrm{SACP}\) or SACP for short) for triplets \((X, V,\mathfrak{F})\), where \(X\) is a Banach space, \(V\) is a \(\tau\)-closed subset of \(X\), \(\mathfrak{F}\) is a subfamily of closed and bounded subsets of \(X\), \(\mathscr{F}\) is a collection of functions, and \(\tau\) is the norm or weak topology on \(X\). We characterize reflexive spaces with the Kadec–Klee property using triplets with \(\tau\)-\(\mathscr{F}\)-\(\mathrm{SACP}\). We investigate the relationship between \(\tau\)-\(\mathscr{F}\)-\(\mathrm{SACP}\) and the continuity properties of the restricted \(f\)-center map. The study further examines \(\tau\)-\(\mathscr{F}\)-\(\mathrm{SACP}\) in the context of \(\mathrm{CLUR}\)-spaces and explores various characterizations of \(\tau\)-\(\mathscr{F}\)-\(\mathrm{SACP}\), including connections to reflexivity, Fréchet smoothness, and the Kadec–Klee property.