Linear Structures in the Set of Norm-Attaining Functionals on a Banach Space

We show, among other results, that if the unit ball of the dual of a Banach space X is $w^*$-sequentially compact, the set of norm-attaining functionals contains a separable norm closed subspace M if and only if the dual $M^*$ of M is the canonical quotient of X. We provide examples of spaces which cannot be renormed in such a way that the set of norm-attaining functionals become a linear space.