Banach空间上的非线性ε-等距

Abstract

令$X$，$Y$为实Banach空间，$\varepsilon\geq0$。映射$f: X\rightarrowY$称为$\varepsilon$-等距， 如果$\big|\|f(x)-f(y)\|-\|x-y\|\big|\leq\varepsilon$，$x,y\inX$。 本文主要研究Banach空间中的凸性、光滑性和基序列在非线性非满$\varepsilon$-等距扰动下的稳定性以及在该扰动下Banach空间$X$，对偶空间$X^*$和二次对偶空间$X^{**}$的内射性、等势内射性和可分内射性的量化特征。同时我们也讨论了Qian的问题在$\mathcal{L}_\infty...

This dissertation mainly studies the rotundity and smoothness of Banach spaces and the stability of basic sequences under the perturbation by nonlinear non-surjective $\varepsilon$-isometries, and characterizes injectivity, separable injectivity and cardinality injectivity of a Banach space $X$, its dual $X^*$ and its second dual $X^{**}$. We also discuss the weak and strong solutions to Qian's p...