This is an announcement for the paper "Strong peak points and denseness of strong peak functions" by Han Ju Lee.
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.
Archive classification: math.FA
Mathematics Subject Classification: 46B04, 46G20, 46G25, 46B22
The source file(s), variationalmethod-2007-04-15.tex: 25864 bytes, is(are) stored in gzipped form as 0705.2650.gz with size 8kb. The corresponding postcript file has gzipped size 75kb.
Submitted from: hahnju@postech.ac.kr
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