Strong Peak Points and Denseness of Strong Peak Functions

Let Cb(K) be the set of all bounded continuous (real or complex) functions on a complete metric space K and A a closed subspace of Cb(K). Using the variational method, it is shown that the set of all strong peak functions in A is dense if and only if the set of all strong peak points is a norming subset of A. As a corollary we show that if X is a locally uniformly convex, complex Banach space, then the set of all strong peak functions in A(BX) is a dense Gδ subset. Moreover if X is separable, smooth and locally uniformly convex, then the set of all norm and numerical strong peak functions in Au(BX : X) is a dense Gδ subset. In case that a set of uniformly strongly exposed points of a (real or complex) Banach space X is a norming subset of P(X) for some n ≥ 1, then the set of all strongly norm attaining elements in P(X) is dense, in particular, the set of all points at which the norm of P(X) is Fréchet differentiable is a dense Gδ subset.