2007/08/30 by Yun Sung Choi, Choi, Yun Sung, Han Ju Lee +3
Mathematics · #46B04 #46B22 #46G20 #46G25 #Advanced Banach Space Theory #FOS: Mathematics #Functional Analysis (math.FA) #Holomorphic and Operator Theory #Nonlinear Differential Equations Analysis #math.FA #msc:46B04 #msc:46B22 #msc:46G20 #msc:46G25
paper · pdf · doi:10.48550/arxiv.0708.4069
arxiv created 2007/08/30 · openalex publication_date 2007/08/30 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 Fréchet 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 strong peak point 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 Ab(BX)=\f∈ Cb(BX) : f|BX^∘ is holomorphic\ is dense. As an application, we show that the smallest closed norming subset of Ab(BX) is the closure of the set of all strong peak points for Ab(BX). This implies that the norm of Ab(BX) is Gâteaux differentiable on a dense subset of Ab(BX), 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 numerical Shilov boundary.