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,…,n}, there exists a closed infinite-dimensional Banach subspace of the space of n-linear forms on ℓ1 such that, for all nonzero elements B of such a subspace, the Arens extension associated to the permutation σ of B is norm-attaining if and only if σ is an element of P. We also study the structure of the set of norm-attaining n-linear forms on c0.