# 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 *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*).

## 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