光滑型的Bishop-Phelps-Bollob\'{a}s定理

Abstract

著名的~Bishop-Phelps~定理~(\cite{34})~是说每个~Banach~空间上的范数可达线性泛函在其对偶空间中总是稠密的.以这一定理为出发点发展起来的变分原理(例如~Ekeland~变分原理~(\cite{16，18})~等)及其应用，已经成为许多现代数学与应用数学分支的基石.~50~多年来，它的各种形式的定量化表述也成为一个新的数学研究领域.然而，在“光滑”情形下的更加精准的论述却没能引起应有的关注.\; 对此，本文引入了“球相对光滑”的概念，并证明了包括所有可分~Banach~空间在内的一类~Banach~空间上总成立如下形式的光滑型量化的~Bishop-Phelp...

The celebrated Bishop-Phelps theorem~(\cite{34}) states that the set of norm attaining~~ functionals on a Banach space is norm dense in the dual space. The variational principles (such as Ekeland variational principle~(\cite{16,18})~etc.) and its applications based on this theorem have become the cornerstone of many branches of modern Mathematics and Applied Mathematrics. For more than 50 years, v...