The Bishop-Phelps-Bollob\'as moduli of a Banach space

TL;DR

This paper introduces two measures called Bishop-Phelps-Bollobás moduli for Banach spaces, establishing bounds, calculating values for specific spaces, and exploring their geometric implications.

Contribution

It defines new moduli to quantify Bishop-Phelps-Bollobás properties, proves bounds, continuity, duality relations, and characterizes spaces with maximal moduli.

Findings

A common upper bound for the moduli across all Banach spaces.

Explicit calculation of the moduli for Hilbert spaces.

Identification of spaces with maximal moduli and their geometric properties.

Abstract

We introduce two Bishop-Phelps-Bollob\'as moduli which measure, for a given Banach space, what is the best possible Bishop-Phelps-Bollob\'as theorem in this space. We show that there is a common upper bound for these moduli for all Banach spaces and we present an example showing that this bound is sharp. We prove the continuity of these moduli and an inequality with respect to duality. We calculate the two moduli for Hilbert spaces and also present many examples for which the moduli have the maximum possible value (among them, there are $C(K)$ spaces and $L_1(\mu)$ spaces). Finally, we show that if a Banach space has the maximum possible value of any of the moduli, then it contains almost isometric copies of the real space $\ell_\infty^{(2)}$ and present an example showing that this condition is not sufficient.