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 language | English |
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Pages (from-to) | 171-182 |
Number of pages | 12 |
Journal | Publications of the Research Institute for Mathematical Sciences |
Volume | 46 |
Issue number | 1 |
DOIs | |
State | Published - 2010 |
Keywords
- Homogeneous polynomial
- Norm-attaining element
- Strong peak point
- Uniformly strongly exposed point