A basis of $\R ^n$ with good isometric properties and some applications to denseness of norm attaining operators

Abstract: We characterize real Banach spaces $Y$ such that the pair $(\ell_\infty ^n, Y)$ has the Bishop-Phelps-Bollob\'as property for operators. To this purpose it is essential the use of an appropriate basis of the domain space $\R^n$. As a consequence of the mentioned characterization, we provide examples of spaces $Y$ satisfying such property. For instance, finite-dimensional spaces, uniformly convex spaces, uniform algebras and $L_1(\mu)$ ($\mu$ a positive measure) satisfy the previous property.