Bishop’s theorem and differentiability of a subspace of C b (K)

Abstract

Let K be a Hausdorff space and Cb(K) be the Banach algebra of all complex bounded continuous functions on K. We study the Gâteaux and Frechet differentiability of subspaces of Cb(K). Using this, we show that the set of all strong peak functions in a nontrivial separating separable subspace H of Cb(K) is a dense Gδ subset of H, if K is compact. This gives a generalized Bishop’s theorem, which says that the closure of the set of all strong peak points for H is the smallest closed norming subset of H. The classical Bishop’s theorem was proved for a separating subalgebra H and a metrizable compact space K.