Banach空间的Lipschitz对偶及其应用

A Novel Dual Notion of a Banach Space: Lipschitz Dual Space

本文引进Banach空间E的一个全新对偶空间概念—Lipschitz对偶空间，并证明：任何Banach空间的Lipschitz对偶空间是某个包含E的Banach空间的线性对偶空间,以所引进的新对偶空间为框架，本文定义了非线性Lipschitz算子的Lipshitz对偶算子，证明：任何非线性Lipschitz算子的Lipschitz对偶算子是有界线性算子．所获结果为推广线性算子理论到非线性情形（特别，运用线性算子理论研究非线性算子的特性）开辟了一条新的途径．作为例证，我们应用所建立的理论证明了若干新的非线性一致Lipschitz映象遍历收敛性定理.

A new dual space notion of a Banach space, named Lipschitz dual space,is introduced, and within the new introduced space framework, the concept of the Lipschitz dual operator of a nonlinear Lipschitz operator is further defined. It is proved that the Lipschitz dual space of any Banach space E is an ordinary dual space of a certain Banach space containing E in the isometric embeding sense, and that the Lipschitz dual operator of any nonlinear Lipscitz operator becomes linear and bounded.By means of these findings, a lot of important results in linear analysis and theorems on linear operators are generalized to nonlinear cases. Thereby, a completely new way to generalize the linear operator theory to the nonlinear case is developed. As examples,several new mean ergodic theorems of the uniformly Lipschitz operators are proved.