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A study on $\mr{F}$-simultaneous approximative $τ$-compactness property in Banach spaces

Syamantak Das, Tanmoy Paul

TL;DR

This work develops and analyzes the $\tau$-$\mathscr{F}$-SACP framework for triplets $(X,V,\mathfrak{F})$ in Banach spaces, introducing core tools $r_f$, $\mathrm{rad}_V^f(F)$, and $\mathrm{Cent}_V^f(F)$ to study simultaneous approximative compactness via restricted $f$-centers. It establishes equivalences between $\tau$-$\mathscr{F}$-SACP and restricted $\mathscr{F}$-center properties, connects these notions to reflexivity and the Kadec–Klee property (including weak forms), and explores how smoothness and rotundity drive the existence and uniqueness of centers, particularly in dual and vector-valued settings. The paper also extends these ideas to $L_p$-type spaces, examines the transfer of Fréchet smoothness to product constructions $\ell_\gamma^N(X)$, and studies the continuity properties of associated set-valued maps, including Lipschitz and upper semicontinuity results and relevant counterexamples. Overall, it provides a cohesive framework linking approximative compactness, Chebyshev centers, and geometric properties of Banach spaces, with implications for the stability of center maps and proximal structures in functional analysis.

Abstract

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

A study on $\mr{F}$-simultaneous approximative $τ$-compactness property in Banach spaces

TL;DR

This work develops and analyzes the --SACP framework for triplets in Banach spaces, introducing core tools , , and to study simultaneous approximative compactness via restricted -centers. It establishes equivalences between --SACP and restricted -center properties, connects these notions to reflexivity and the Kadec–Klee property (including weak forms), and explores how smoothness and rotundity drive the existence and uniqueness of centers, particularly in dual and vector-valued settings. The paper also extends these ideas to -type spaces, examines the transfer of Fréchet smoothness to product constructions , and studies the continuity properties of associated set-valued maps, including Lipschitz and upper semicontinuity results and relevant counterexamples. Overall, it provides a cohesive framework linking approximative compactness, Chebyshev centers, and geometric properties of Banach spaces, with implications for the stability of center maps and proximal structures in functional analysis.

Abstract

Veselý (1997) studied Banach spaces that admit -centers for finite subsets of the space. In this work, we introduce the concept of -simultaneous approximative -compactness property (--SACP in short) for triplets , where is a Banach space, is a -closed subset of , is a subfamily of closed and bounded subsets of , is a collection of functions, and is the norm or weak topology on . We characterize reflexive spaces with the Kadec-Klee property using triplets with --SACP. We investigate the relationship between --SACP and the continuity properties of the restricted -center map. The study further examines --SACP in the context of spaces and explores various characterizations of --SACP, including connections to reflexivity, Fréchet smoothness, and the Kadec-Klee property.
Paper Structure (6 sections, 21 theorems, 15 equations)

This paper contains 6 sections, 21 theorems, 15 equations.

Table of Contents

  1. Introduction
  2. Prerequisites:
  3. Observations:
  4. Various characterizations of $\tau$-$\mathscr{F}$-SACP
  5. Various implications of smoothness and rotundity in relation to $\tau$-$\mathscr{F}$-SACP
  6. Continuity properties of some set-valued maps in relation to $\tau$-$\mathscr{F}$-SACP

Key Result

Theorem 2.2

Let $V$ be a $\tau$-closed subset of $X$ and $\mathcal{A}\subseteq \mathcal{F}(X)$. Then, $(X,V,\mathcal{A})$ has $\tau$-$\mathscr{F}_{cm}$-$\bf{\emph{SACP}}$ if and only if $(X,V,\mathcal{A})$ has the $\tau$-$\mathscr{F}_{cm}$-property-$(P_1)$ and $\emph{Cent}_V^f(F)$ is $\tau$-compact for all $F\i

Theorems & Definitions (47)

  • Definition 1.1
  • Definition 1.2
  • Definition 1.3
  • Definition 1.4
  • Definition 1.5
  • Definition 1.6
  • Definition 1.7
  • Example 2.1
  • Theorem 2.2
  • proof
  • ...and 37 more