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EQUIVALENT NORMS WITH AN EXTREMELY NONLINEABLE SET OF NORM ATTAINING FUNCTIONALS
- Part of
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- Journal:
- Journal of the Institute of Mathematics of Jussieu / Volume 19 / Issue 1 / January 2020
- Published online by Cambridge University Press:
- 13 February 2018, pp. 259-279
- Print publication:
- January 2020
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- Article
- Export citation
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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.
