On proximinality of subspaces and the lineability of the set of norm-attaining functionals of Banach spaces

TL;DR

This paper constructs specific Banach spaces demonstrating precise proximinal subspace properties and the structure of norm-attaining functionals, addressing longstanding problems in Banach space theory.

Contribution

It provides explicit examples of Banach spaces with controlled proximinal subspace dimensions and norm-attaining functional structures, extending previous results by Read and Rmoutil.

Findings

Existence of Banach spaces with proximinal subspaces of certain codimensions but not higher.

Construction of spaces where the set of norm-attaining functionals contains specific dimensional subspaces.

Identification of non-separable Banach spaces with norm-attaining functionals containing infinite-dimensional separable subspaces but no non-separable ones.

Abstract

We show that for every $1<n<\infty$, there exits a Banach space $X_n$ containing proximinal subspaces of codimension $n$ but no proximinal finite codimensional subspaces of higher codimension. Moreover, the set of norm-attaining functionals of $X_n$ contains $n$-dimensional subspaces, but no subspace of higher dimension. This gives a $n$-by-$n$ version of the solutions given by Read and Rmoutil to problems of Singer and Godefroy. We also study the existence of strongly proximinal subspaces of finite codimension, showing that for every $1<n<\infty$ and $1\leqslant k <n$, there is a Banach space $X_{n,k}$ containing proximinal subspaces of finite codimension up to $n$ but not higher, and containing strongly proximinal subspaces of finite codimension up to $k$ but not higher. Finally, we deal with possible infinite-dimensional versions of the previous results, showing that there are \emph{non-separable} Banach spaces whose set of norm-attaining functionals contains infinite-dimensional separable linear subspaces but no non-separable subspaces.