On Universally Left-stability of ε-Isometry  

作　　者：Ling Xin BAO Li Xin CHENG Qing Jin CHENG Duan Xu DAI 

机构地区：[1]School of Mathematical Sciences,Xiamen University

出　　处：《Acta Mathematica Sinica,English Series》2013年第11期2037-2046,共10页数学学报（英文版）

基　　金：Supported by National Natural Science Foundation of China(Grant Nos.11071201 and 11001231)

摘　　要：Let X, Y be two real Banach spaces and ε≥0. A map f ： X → Y is said to be a standard ε-isometry if│││f/（x） - f（y）││ - ]ix - Y││x-y││ ε for all x,y C X and with f（O） = O. We say that a pair of Banach spaces （X, Y） is stable if there exists γ〉 0 such that, for every such ε and every standard v-isometry f ： X → Y, there is a bounded linear operator T ： L（f） → f（X） → X so that ││Tf（x） - x││ ≤γε for all x E X. X（Y） is said to be universally left-stable if （X, Y） is always stable for every Y（X）. In this paper, we show that if a dual Banach space X is universally left-stable, then it is isometric to a complemented w＊-closed subspace of ∞ （1） for some set F, hence, an injective space; and that a Banach space is universally left-stable if and only if it is a cardinality injective space; and universally left-stability spaces are invariant.Let X, Y be two real Banach spaces and ε≥0. A map f ： X → Y is said to be a standard ε-isometry if│││f/（x） - f（y）││ - ]ix - Y││x-y││ ε for all x,y C X and with f（O） = O. We say that a pair of Banach spaces （X, Y） is stable if there exists γ〉 0 such that, for every such ε and every standard v-isometry f ： X → Y, there is a bounded linear operator T ： L（f） → f（X） → X so that ││Tf（x） - x││ ≤γε for all x E X. X（Y） is said to be universally left-stable if （X, Y） is always stable for every Y（X）. In this paper, we show that if a dual Banach space X is universally left-stable, then it is isometric to a complemented w＊-closed subspace of ∞ （1） for some set F, hence, an injective space; and that a Banach space is universally left-stable if and only if it is a cardinality injective space; and universally left-stability spaces are invariant.

关 键 词：ε-Isometry linear isometry stability injective space Banach space 

分 类 号：O177.2[理学—数学]