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 $B_X$ 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 $f{|}_{B_X^{\circ }}$ 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 $A_b(B_X)$.