Denseness of Norm-Attaining Mappings on Banach Spaces

Abstract

Let X and Y be Banach spaces. Let P(X-n : 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(X-n : Y) when a set of use. points of the unit ball B-X is dense in the unit sphere S-X. 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 A(b)(B-X : Y) be the Banach space of all bounded continuous Y-valued mappings f on B-X whose restrictions f vertical bar(B)degrees(X) to the open unit, ball are holomorphic. It follows that the set of all norm-attaining elements is dense in A(b)(B-X : Y) if the set of all strong peak points in A(b)(B-X) is a norming subset for A(b)(B-X).