Denseness of norm-attaining mappings on banach spaces

Yun Sung Choi, Han Ju Lee, Hyun Gwi Song

Research output: Contribution to journalArticlepeer-review

6 Scopus citations

Abstract

Let X and Y be Banach spaces. Let P(nX : 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(nX : 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 Cb[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 B X 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 A b(BX) is a norming subset for Ab(B X).

Original languageEnglish
Pages (from-to)171-182
Number of pages12
JournalPublications of the Research Institute for Mathematical Sciences
Volume46
Issue number1
DOIs
StatePublished - 2010

Keywords

  • Homogeneous polynomial
  • Norm-attaining element
  • Strong peak point
  • Uniformly strongly exposed point

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