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Abstract

The Bishop–Phelps Theorem states the denseness of the set of norm attaining functionals in the topological dual of any Banach space. Bollobás obtained a quantitative version of this result showing that a pair $(x, x^*)$ given by an element $x$ in the unit sphere of a Banach space and a functional $x^*$ in the unit sphere of the dual such that $x^* (x)$ is close to $1$ can be approximated in norm by another pair $(y,y^*)$ satisfying the same conditions and also that $y^*$ attains its norm at $y$. In 2008 the problem of obtaining versions of Bollobás result for operators was considered. In this survey we include most of the results on this topic to the present day.