Bishop's Theorem and Differentiability of a subspace of $C_b(K)$
| dc.creator | Choi, Yun Sung | |
| dc.creator | Lee, Han Ju | |
| dc.creator | Song, Hyun Gwi | |
| dc.date | 2007-08-30 | |
| dc.date.accessioned | 2026-07-07T08:26:38Z | |
| dc.date.available | 2026-07-07T08:26:38Z | |
| dc.description | 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 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 $A_b(B_X)=\{f\in C_b(B_X) : f|_{B_X^\circ} {is holomorphic}\}$ 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 numerical Shilov boundary. | |
| dc.identifier | https://arxiv.org/abs/0708.4069 | |
| dc.identifier | http://arxiv.org/abs/0708.4069 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/137005 | |
| dc.subject | Functional Analysis | |
| dc.subject | 46B04; 46G20; 46G25; 46B22 | |
| dc.title | Bishop's Theorem and Differentiability of a subspace of $C_b(K)$ | |
| dc.type | text |