The Bishop–Phelps–Bollobás theorem on bounded closed convex sets

This paper deals with the Bishop–Phelps–Bollobás 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 ( β ) , 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 ∞ ‐sum of a family of Banach spaces.