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Functions attaining the supremum and isomorphic properties of a Banach space
Mar\'{\i}a D. Acosta,Julio Becerra Guerrero,Manuel Ruiz 대한수학회 2004 대한수학회지 Vol.41 No.1
We prove that a Banach space that is convex-transitive and such that for some element u in the unit sphere, and for every subspace M containing u, it happens that the subset of norm attaining functionals on M is second Baire category in M∗ is, in fact, almost-transitive and superreflexive. We also obtain a characterization of finite-dimensional spaces in terms of functions that attain their supremum: a Banach space is finite-dimensional if, for every equivalent norm, every rank-one operator attains its numerical radius. Finally, we describe the subset of norm attaining functionals on a space isomorphic to 1, where the norm is the restriction of a Luxembourg norm on L1. In fact, the subset of norm attaining functionals for this norm coincides with the subset of norm attaining functionals for the usual norm.