Norm attaining operators and variational principle

Abstract

We establish a linear variational principle extending the Deville-Godefroy-Zizler's one. We use this variational principle to prove that if $X$ is a Banach space having property $(\alpha )$ of Schachermayer and $Y$ is any banach space, then the set of all norm strongly attaining linear operators from $X$ into $Y$ is a complement of a $\sigma$-porous set. Moreover, the results of the paper applies also to an abstract class of (linear and nonlinear) operator spaces.