# Denseness of Norm-Attaining Mappings on Banach Spaces

### Yun Sung Choi

Pohang University of Science and Technology, Pohang City, Kyungbuk, South Korea### Han Ju Lee

Dongguk University, Seoul, South Korea### Hyun Gwi Song

Sogang University, Seoul, South Korea

## 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 $f∣_{B_{X}}$ 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 $A_{b}(B_{X})$.

## Cite this article

Yun Sung Choi, Han Ju Lee, Hyun Gwi Song, Denseness of Norm-Attaining Mappings on Banach Spaces. Publ. Res. Inst. Math. Sci. 46 (2010), no. 1, pp. 171–182

DOI 10.2977/PRIMS/4