Two refinements of the Bishop-Phelps-Bollob\'as modulus

TL;DR

This paper refines the Bishop-Phelps-Bollobás theorem by providing sharp estimates of the associated modulus, relating it to geometric properties of Banach spaces, and calculating it explicitly in specific cases like Hilbert spaces.

Contribution

It offers a sharp general estimation of the Bishop-Phelps-Bollobás modulus and links it to the space's geometric property of uniform non-squareness, with explicit calculations for certain spaces.

Findings

Sharp estimation of the Bishop-Phelps-Bollobás modulus as a function of norms.

Explicit calculation of the modulus in Hilbert spaces.

Uniformly non-square spaces cannot attain the maximum modulus.

Abstract

Extending the celebrated result by Bishop and Phelps that the set of norm attaining functionals is always dense in the topological dual of a Banach space, Bollob\'as proved the nowadays known as the Bishop-Phelps-Bollob\'as theorem, which allows to approximate at the same time a functional and a vector in which it almost attains the norm. Very recently, two Bishop-Phelps-Bollob\'as moduli of a Banach space have been introduced [J. Math. Anal. Appl. 412 (2014), 697--719] to measure, for a given Banach space, what is the best possible Bishop-Phelps-Bollob\'as theorem in this space. In this paper we present two refinements of the results of that paper. On the one hand, we get a sharp general estimation of the Bishop-Phelps-Bollob\'as modulus as a function of the norms of the point and the functional, and we also calculate it in some examples, including Hilbert spaces. On the other hand, we relate the modulus of uniform non-squareness with the Bishop-Phelps-Bollob\'as modulus obtaining, in particular, a simpler and quantitative proof of the fact that a uniformly non-square Banach space cannot have the maximum value of the Bishop-Phelps-Bollob\'as modulus.