Uniform Convexity and the Bishop–Phelps–Bollobás Property

Abstract

Abstract A new characterization of the uniform convexity of Banach space is obtained in the sense of the Bishop–Phelps–Bollobás theorem. It is also proved that the couple of Banach spaces (X;Y) has the Bishop–Phelps–Bollobás property for every Banach space Y when X is uniformly convex. As a corollary, we show that the Bishop–Phelps–Bollobás theorem holds for bilinear forms on ℓ p × ℓ q (1 < p; q < ∞).