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EQUIVALENT NORMS WITH AN EXTREMELY NONLINEABLE SET OF NORM ATTAINING FUNCTIONALS

  • Vladimir Kadets (a1), Ginés López (a2), Miguel Martín (a2) and Dirk Werner (a3)

Abstract

We present a construction that enables one to find Banach spaces $X$ whose sets $\operatorname{NA}(X)$ of norm attaining functionals do not contain two-dimensional subspaces and such that, consequently, $X$ does not contain proximinal subspaces of finite codimension greater than one, extending the results recently provided by Read [Banach spaces with no proximinal subspaces of codimension 2, Israel J. Math. (to appear)] and Rmoutil [Norm-attaining functionals need not contain 2-dimensional subspaces, J. Funct. Anal. 272 (2017), 918–928]. Roughly speaking, we construct an equivalent renorming with the requested properties for every Banach space $X$ where the set $\operatorname{NA}(X)$ for the original norm is not “too large”. The construction can be applied to every Banach space containing $c_{0}$ and having a countable system of norming functionals, in particular, to separable Banach spaces containing $c_{0}$ . We also provide some geometric properties of the norms we have constructed.

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EQUIVALENT NORMS WITH AN EXTREMELY NONLINEABLE SET OF NORM ATTAINING FUNCTIONALS

  • Vladimir Kadets (a1), Ginés López (a2), Miguel Martín (a2) and Dirk Werner (a3)

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