Article ID | Journal | Published Year | Pages | File Type |
---|---|---|---|---|
8896787 | Journal of Functional Analysis | 2018 | 11 Pages |
Abstract
We study geometric properties of the Banach space R constructed recently by C. Read [8] which does not contain proximinal subspaces of finite codimension greater than or equal to two. Concretely, we show that the bidual of R is strictly convex, that the norm of the dual of R is rough, and that R is weakly locally uniformly rotund (but it is not locally uniformly rotund). Apart of the own interest of the results, they provide a simplification of the proof by M. Rmoutil [9] that the set of norm-attaining functionals over R does not contain any linear subspace of dimension greater than or equal to two. Note that if a Banach space X contains proximinal subspaces of finite codimension at least two, then the set of norm-attaining functionals over X contain two-dimensional linear subspaces of Xâ. Our results also provides positive answer to the questions of whether the dual of R is smooth and of whether R is weakly locally uniformly rotund [9]. Finally, we present a renorming of Read's space which is smooth, whose dual is smooth, and such that its set of norm-attaining functionals does not contain any linear subspace of dimension greater than or equal to two, so the renormed space does not contain proximinal subspaces of finite codimension greater than or equal to two.
Related Topics
Physical Sciences and Engineering
Mathematics
Algebra and Number Theory
Authors
Vladimir Kadets, Ginés López-Pérez, Miguel MartÃn,