Denseness of Norm-Attaining Mappings on Banach Spaces
Yun Sung Choi
Pohang University of Science and Technology, Pohang City, Kyungbuk, South KoreaHan Ju Lee
Dongguk University, Seoul, South KoreaHyun 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