Strong peak points and denseness of strong peak functions

Let $C_b(K)$ be the set of all bounded continuous (real or complex) functions on a complete metric space $K$ and $A$ a closed subspace of $C_b(K)$. Using the variational method, it is shown that the set of all strong peak functions in $A$ is dense if and only if the set of all strong peak points is a norming subset of $A$. As a corollary we show that if $X$ is a locally uniformly convex, complex Banach space, then the set of all strong peak functions in $\mathcal{A}(B_X)$ is a dense $G_{\delta }$ subset. Moreover if $X$ is separable, smooth and locally uniformly convex, then the set of all norm and numerical strong peak functions in ${\mathcal{A}}_u(B_X:X)$ is a dense $G_{\delta }$ subset. In case that a set of uniformly strongly exposed points of a (real or complex) Banach space $X$ is a norming subset of $\mathcal{P}({}^{n}X)$ for some $n\ge 1$,thenthesetofallstronglynormattainingelementsin$$\mathcal{P}$({}^n X)$isdense,inparticular,thesetofallpointsatwhichthenormof$$\mathcal{P}$({}^n X)$isFr\acute{e}chetdifferentiableisadense$G_\delta$ subset.