This is an announcement for the paper "Norm-attaining functionals and proximinal subspaces" by Martin Rmoutil.
Abstract: G. Godefroy asked whether, on any Banach space, the set of norm-attaining functionals contains a 2-dimensional linear subspace. We prove that a recent construction due to C.J. Read provides an example of a space which does not have this property. This is done through a study of the relation between the following two sentences where X is a Banach space and Y is a closed subspace of finite codimension in X: (A) Y is proximinal in X. (B) The annihilator of Y consists of norm-attaining functionals. We prove that these are equivalent if X is the Read's space. Moreover, we prove that any non-reflexive Banach space X with any given closed subspace Y of finite codimension at least 2 admits an equivalent norm such that (B) is true and (A) is false.
Archive classification: math.FA
Mathematics Subject Classification: 46B10, 46B20, 46B03
Submitted from: martin@rmoutil.eu
The paper may be downloaded from the archive by web browser from URL
http://front.math.ucdavis.edu/1503.06112
or
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alspach＠math.okstate.edu