The Bishop-Phelps-Bollobas theorem on bounded closed convex sets

Abstract

This paper deals with the Bishop-Phelps-Bollobas property (BPBP) on bounded closed convex subsets of a Banach space X, not just on its closed unit ball B-X. We prove that BPBP holds for bounded linear functionals on arbitrary bounded closed convex subsets of a real Banach space. We show that, for a Banach space Y with property (beta), the pair (X, Y) has 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 a positive modulus of convexity, we show that the pair (X, Y) has BPBP on D for every Banach space Y. We further obtain that, for an Asplund space X and for a locally compact Hausdorff space L, the pair (X, C-0(L)) has BPBP on every bounded closed absolutely convex subset D of X. Finally, we study the stability of BPBP on a bounded closed convex set for the l(1)-sum or l(infinity)-sum of a family of Banach spaces.