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We show that for every $1<n<\infty$, there exits a Banach space $X_n$containing proximinal subspaces of codimension $n$ but no proximinal finitecodimensional subspaces of higher codimension. Moreover, the set ofnorm-attaining functionals of $X_n$ contains $n$-dimensional subspaces, but nosubspace of higher dimension. This gives a $n$-by-$n$ version of the solutionsgiven by Read and Rmoutil to problems of Singer and Godefroy. We also study theexistence of strongly proximinal subspaces of finite codimension, showing thatfor 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 ofthe previous results, showing that there are \emph{non-separable} Banach spaceswhose set of norm-attaining functionals contains infinite-dimensional separablelinear subspaces but no non-separable subspaces.

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