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).
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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. 171–182DOI 10.2977/PRIMS/4