Norming points and critical points

Using a diffeomorphism between the unit sphere and a closed hyperplane of an infinite dimensional Banach space, we introduce the differentiation of a function defined on the unit sphere, and show that a continuous linear functional attains its norm if and only if it has a critical point on the unit sphere. Furthermore, we provide a strong version of the Bishop–Phelps–Bollobás theorem for a Lipschitz smooth Banach space.