The Bishop-Phelps Theorem

Abstract

This chapter outlines the history of the Bishop–Phelps theorem. This theorem is an elementary but fundamental result about convex sets and continuous linear functional on real Banach spaces. The chapter presents the ideas underlying the proof of the original result. At a 1985 conference at Kent State University, Gilles Godefroy raised the question as to whether there is a valid version of the Bishop–Phelps theorem in complex Banach spaces. The answer is trivially “yes” if a person restricts himself to the real scalars and the real parts of linear functional. The question was raised in whether linear operators T that attain their norms are necessarily dense in the space of all bounded operators from one Banach space to another. The question as to whether vector-valued lower semicontinuous convex functions need to have subdifferentials was answered in the negative way; this is a consequence of an example of two proper lower semi-continuous functions on l2 with no common point of subdifferentiability.