COARSE ISOMETRIES BETWEEN FINITE DIMENSIONAL BANACH SPACES

doi:10.1007/s10473-021-0506-5

数学物理学报(英文版) ›› 2021, Vol. 41 ›› Issue (5): 1493-1502.doi: 10.1007/s10473-021-0506-5

COARSE ISOMETRIES BETWEEN FINITE DIMENSIONAL BANACH SPACES

孙玉奇, 张文

School of Mathematical Science, Xiamen University, Xiamen 361005, China

收稿日期:2020-05-20 修回日期:2021-04-29 出版日期:2021-10-25 发布日期:2021-10-21
通讯作者: Wen ZHANG E-mail:wenzhang@xmu.edu.cn
作者简介:Yuqi SUN,E-mail:sunyuqi00@163.com
基金资助:
Supported by National Natural Science Foundation of China (11731010 and 12071388).

COARSE ISOMETRIES BETWEEN FINITE DIMENSIONAL BANACH SPACES

Yuqi SUN, Wen ZHANG

School of Mathematical Science, Xiamen University, Xiamen 361005, China

Received:2020-05-20 Revised:2021-04-29 Online:2021-10-25 Published:2021-10-21
Contact: Wen ZHANG E-mail:wenzhang@xmu.edu.cn
Supported by:
Supported by National Natural Science Foundation of China (11731010 and 12071388).

摘要/Abstract

摘要： Assume that $X$ and $Y$ are real Banach spaces with the same finite dimension. In this paper we show that if a standard coarse isometry $f:X\rightarrow Y$ satisfies an integral convergence condition or weak stability on a basis, then there exists a surjective linear isometry $U:X\rightarrow Y$ such that $\|f(x)-Ux\|=o(\|x\|)$ as $\|x\|\rightarrow\infty$. This is a generalization about the result of Lindenstrauss and Szankowski on the same finite dimensional Banach spaces without the assumption of surjectivity. As a consequence, we also obtain a stability result for $\varepsilon$-isometries which was established by Dilworth.

关键词: coarse isometry, linear isometry, finite dimensional Banach spaces

Abstract: Assume that $X$ and $Y$ are real Banach spaces with the same finite dimension. In this paper we show that if a standard coarse isometry $f:X\rightarrow Y$ satisfies an integral convergence condition or weak stability on a basis, then there exists a surjective linear isometry $U:X\rightarrow Y$ such that $\|f(x)-Ux\|=o(\|x\|)$ as $\|x\|\rightarrow\infty$. This is a generalization about the result of Lindenstrauss and Szankowski on the same finite dimensional Banach spaces without the assumption of surjectivity. As a consequence, we also obtain a stability result for $\varepsilon$-isometries which was established by Dilworth.

Key words: coarse isometry, linear isometry, finite dimensional Banach spaces

中图分类号:

46B04

孙玉奇, 张文. COARSE ISOMETRIES BETWEEN FINITE DIMENSIONAL BANACH SPACES[J]. 数学物理学报(英文版), 2021, 41(5): 1493-1502.

Yuqi SUN, Wen ZHANG. COARSE ISOMETRIES BETWEEN FINITE DIMENSIONAL BANACH SPACES[J]. Acta mathematica scientia,Series B, 2021, 41(5): 1493-1502.

参考文献

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COARSE ISOMETRIES BETWEEN FINITE DIMENSIONAL BANACH SPACES

COARSE ISOMETRIES BETWEEN FINITE DIMENSIONAL BANACH SPACES

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