关于映射一致可微性的几个定理

一致可微是分析学中的重点与难点,以往学界多从一维情形讨论其充要条件,文章将其推广到高维情形,证明了映射一致可微当且仅当映射的微分算子即矩阵算子在算子范数的意义下一致连续;同时给出判定矩阵算子一致连续的充要条件,即矩阵算子里的每一个元素一致连续.在此基础上,进一步考虑无穷维空间的一致可微,证明了当映射在紧集的ε0-邻域上C1时,则映射在紧集的δ1(

Some theorems on uniform differentiability of mapping

It is important and difficult to study the concept of uniform differentiability in analysis . The sufficient and necessary condition for it is usually discussed in the one-dimension .In this paper ,we extended to the high-dimension ,and showed that the mapping is uniform differential if and only if the differential operator （i .e .,the matrix operator ） of mapping is uniform continuous in the sense of operator norm .Meanwhile ,we also gave a equivalent condition for judging the uniform continuous to matrix operator ,i .e .,every element of matrix is uniform continuous .Furthermore ,we considered it in infinite dimensional ,and proved that if the mapping is C1 in ε0 -neighborhood of compact set ,it will be uniform differential in a δ1 （〈ε0 ）-neighborhood .