Bishop's theorem and differentiability of a subspace of C_b(K)

Let K be a Hausdorff space and C_b (K) be the Banach algebra of all complex bounded continuous functions on K. We study the Gâteaux and Fréchet differentiability of subspaces of C_b (K). Using this, we show that the set of all strong peak functions in a nontrivial separating separable subspace H of C_b (K) is a dense G_δ 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 In the case that X is a complex Banach space with the Radon-Nikodým property, we show that the set of all strong peak functions in A_b(B_X)={f ∈ C_b(B_X): f{pipe}_B ^̊ _x is holomorphic} is dense. As an application, we show that the smallest closed norming subset of A_b(B_X) is the closure of the set of all strong peak points for A_b (B_X). This implies that the norm of A_b(B_X) is Gâteaux differentiable on a dense subset of A_b (B_X), even though the norm is nowhere Fréchet differentiable when X is nontrivial. We also study the denseness of norm attaining holomorphic functions and polynomials. Finally we investigate the existence of the numerical Shilov boundary.