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Let $C_b(K)$ be the set of all bounded continuous (real or complex) functionson a complete metric space $K$ and $A$ a closed subspace of $C_b(K)$. Using thevariational 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 normingsubset of $A$. As a corollary we show that if $X$ is a locally uniformlyconvex, 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 numericalstrong 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 $n\ge1$, then the set of all strongly norm attaining elements in $\mathcal{P}({}^nX)$ is dense, in particular, the set of all points at which the norm of$\mathcal{P}({}^n X)$ is Fr\'echet differentiable is a dense $G_\delta$ subset.

Strong peak points and denseness of strong peak functions