Strong peak points and denseness of strong peak functions

TL;DR

This paper investigates the density and denseness of strong peak functions and points in various function spaces on metric and Banach spaces, establishing conditions for their prevalence and differentiability properties.

Contribution

It characterizes when strong peak functions are dense in subspaces of bounded continuous functions and applies these results to Banach spaces with specific convexity and smoothness properties.

Findings

Strong peak functions are dense iff strong peak points form a norming set.

In locally uniformly convex Banach spaces, strong peak functions form a dense G_delta set.

In spaces with strongly exposed points, strongly norm attaining elements are dense.

Abstract

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 $n\ge 1$, then the set of all strongly norm attaining elements in $\mathcal{P}({}^n X)$ 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.