A variational approach to norm attainment of some operators and polynomials

If X is an Asplund space, then every uniformly continuous function on BX* which is holomorphic on the open unit ball, can be perturbed by a w* continuous and homogeneous polynomial on X* to obtain a norm attaining function on the dual unit ball. This is a consequence of a version of Bourgain-Stegall’s variational principle. We also show that the set of N-homogeneous polynomials between two Banach spaces X and Y whose transposes attain their norms is dense in the corresponding space of N-homogeneous polynomials. In the case when Y is the space of Radon measures on a compact K, this result can be strengthened.