Denseness of Norm-Attaining Mappings on Banach Spaces

Abstract

Let X and Y be Banach spaces. Let P(n__X : Y) be the space of all Y-valued continuous n-homogeneous polynomials on X. We show that the set of all norm-attaining elements is dense in P(n__X : Y) when a set of u.s.e. points of the unit ball BX is dense in the unit sphere SX. Applying strong peak points instead of u.s.e. points, we generalize this result to a closed subspace of C_b_(M, Y), where M is a complete metric space. For complex Banach spaces X and Y, let Ab(BX : Y) be the Banach space of all bounded continuous Y-valued mappings f on BX whose restrictions f|_BX_∘ to the open unit ball are holomorphic. It follows that the set of all norm-attaining elements is dense in Ab(BX : Y) if the set of all strong peak points in Ab(BX) is a norming subset for Ab(BX).