BISHOP'S THEOREM AND DIFFERENTIABILITY OF A SUBSPACE OF C-b(K)

Abstract

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 Gateaux and Frechet 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(delta) 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-Nikodym property, we show that the set of all strong peak functions in A(b)(B-X) = {f is an element of C-b(B-X) : f vertical bar B-X degrees 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 Gateaux differentiable on a dense subset of A(b)(B-X), even though the norm is nowhere Frechet 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.