  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 G\^{a}teaux and
Fr\'echet 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 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-Nikod\'ym
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 G\^{a}teaux differentiable on a dense subset of $A_b(B_X)$, even
though the norm is nowhere Fr\'echet 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.
