Spaceability in norm-attaining sets

Abstract

We study the existence of infinite-dimensional vector spaces in the sets of norm-attaining operators, multilinear forms, and polynomials. Our main result is that, for every set of permutations $P$ of the set $\{1,\dots ,n\}$, there exists a closed infinite-dimensional Banach subspace of the space of $n$-linear forms on ${\ell }_1$ such that, for all nonzero elements $B$ of such a subspace, the Arens extension associated to the permutation $\sigma$ of $B$ is norm-attaining if and only if $\sigma$ is an element of $P$. We also study the structure of the set of norm-attaining $n$-linear forms on $c_0$.