值域和Domain为Infinite Dimensional的函数的广义雅可比矩阵
Generalized Jacobian for Functions with Infinite Dimensional Range and Domain
摘要：
本文考虑了作用在无穷维赋范空间之间的局部Lipschitz函数。当值域是对偶空间且满足Radon-Nikodym性质时，Clarke的广义雅可比矩阵将推广到这种情形.给出了广义广义雅可比矩阵的刻画和基本性质，包括非空性、$β$ -紧性、$β$ -上连续性和一个中值定理.与已知的概念提供了连接和链式法则证明使用开发的关键成果。这包括向量化和限制定理，以及扩张定理。从而证明了本文所引入的广义雅可比矩阵具有导似集的所有性质。
英文摘要：
In this paper, locally Lipschitz functions acting between infinite dimensional normed spaces are considered. When the range is a dual space and satisfies the Radon--Nikodým property, Clarke's generalized Jacobian will be extended to this setting. Characterization and fundamental properties of the extended generalized Jacobian are established including the nonemptiness, the $β$-compactness, the $β$-upper semicontinuity, and a mean-value theorem. A connection with known notions is provided and chain rules are proved using key results developed. This included the vectorization and restriction theorem, and the extension theorem. Therefore, the generalized Jacobian introduced in this paper is proved to enjoy all the properties required of a derivative like-set.
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