关于矩阵值Lipschitz映射空间的若干研究

Some Properties of Matrix-range Lipschitz Spaces

引入并研究了由紧的距离空间(K,d)到Mm,n(F)中的Lipschitz-α映射构成的空间Lα(K,Mm,n(F))和lα(K,Mm,n(F));并证明了它们关于范数‖f‖α=‖f‖∞+Lα(f)是Lipschitz空间;得到了lα(K,Mm,n(F))是Lα(K,Mm,n(F))的闭子空间;当0<α≤β≤1时,Lβ(K,Mm,n(F))是Lα(K,Mm,n(F))的闭子空间.

Spaces, L^α（K,Mm,n（F）） and l^α（K,Mm,n（F））, constructed with Lipschitz-α mapped from a compact range space （K, d） to a matrix space Mm,n（F）, are introduced and studied. It is proved that both of them are Lipschitz spaces in the norm ｜｜·｜｜α（｜｜f｜｜α=｜｜f｜｜∞＋Lα（f）） ; l^α（K,Mm,n（F）） is a closed subspace of L^α（K,Mm,n（F）） ; and if 0〈α≤β≤1, then L^β（K,Mm,n（F）） is a closed subspace of L^α（K,Mm,n（F））.