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

Abstract

We show that for every 1<n<∞, there exists a Banach space Xn containing proximinal subspaces of codimension n but no proximinal finite codimensional subspaces of higher codimension. Moreover, the set of norm-attaining functionals of Xn contains n-dimensional subspaces, but no subspace of higher dimension. This gives an 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<∞ and 1⩽k<n, there is a Banach space Xn,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, among other results, that there are non-separable Banach spaces whose set of norm-attaining functionals contains infinite-dimensional separable linear subspaces but no non-separable subspaces.