Bishop-Phelps-Bolloba's theorem on bounded closed convex sets

TL;DR

This paper extends the Bishop-Phelps-Bollobás property to bounded closed convex sets in Banach spaces, providing new conditions and stability results for various classes of Banach spaces and convex subsets.

Contribution

It generalizes BPBp from the unit ball to arbitrary convex subsets and establishes its validity for finite-dimensional, property (β), and Asplund spaces, along with stability results.

Findings

BPBp holds for bounded linear functionals on arbitrary convex subsets.

Finite-dimensional Banach spaces satisfy BPBp on convex subsets.

BPBp is stable under certain sums of Banach spaces.

Abstract

This paper deals with the \emph{Bishop-Phelps-Bollob\'as property} (\emph{BPBp} for short) on bounded closed convex subsets of a Banach space $X$, not just on its closed unit ball $B_X$. We firstly prove that the \emph{BPBp} holds for bounded linear functionals on arbitrary bounded closed convex subsets of a real Banach space. We show that for all finite dimensional Banach spaces $X$ and $Y$ the pair $(X,Y)$ has the \emph{BPBp} on every bounded closed convex subset $D$ of $X$, and also that for a Banach space $Y$ with property $(\beta)$ the pair $(X,Y)$ has the \emph{BPBp} on every bounded closed absolutely convex subset $D$ of an arbitrary Banach space $X$. For a bounded closed absorbing convex subset $D$ of $X$ with positive modulus convexity we get that the pair $(X,Y)$ has the \emph{BPBp} on $D$ for every Banach space $Y$. We further obtain that for an Asplund space $X$ and for a locally compact Hausdorff $L$, the pair $(X, C_0(L))$ has the \emph{BPBp} on every bounded closed absolutely convex subset $D$ of $X$. Finally we study the stability of the \emph{BPBp} on a bounded closed convex set for the $\ell_1$-sum or $\ell_{\infty}$-sum of a family of Banach spaces.