Banach Journal of Mathematical Analysis

On the Aleksandrov-Rassias problem and the Hyers-Ulam-Rassias stability problem

Liyun Tan and Shuhuang Xiang

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Abstract

Let $X$ and $Y$ be normed linear spaces. A mapping $T:X \rightarrow Y$ is called preserving the distance $r$ if for all $x,y$ of $X$ with $\|x-y\|_X=r$ then $\|T(x)-T(y)\|=r$. In this paper, we provide an overall account of the development of the Aleksandrov problem, the Aleksandrov-Rassias problem for mappings which preserve distances and details for the Hyers-Ulam-Rassias stability problem.

Article information

Source
Banach J. Math. Anal., Volume 1, Number 1 (2007), 11-22.

Dates
First available in Project Euclid: 21 April 2009

Permanent link to this document
https://projecteuclid.org/euclid.bjma/1240321551

Digital Object Identifier
doi:10.15352/bjma/1240321551

Mathematical Reviews number (MathSciNet)
MR2350190

Zentralblatt MATH identifier
1130.39027

Subjects
Primary: 39B82: Stability, separation, extension, and related topics [See also 46A22]
Secondary: 44B20 46C05: Hilbert and pre-Hilbert spaces: geometry and topology (including spaces with semidefinite inner product)

Keywords
isometry stability additive mapping conservative distance

Citation

Tan, Liyun; Xiang, Shuhuang. On the Aleksandrov-Rassias problem and the Hyers-Ulam-Rassias stability problem. Banach J. Math. Anal. 1 (2007), no. 1, 11--22. doi:10.15352/bjma/1240321551. https://projecteuclid.org/euclid.bjma/1240321551


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