Injectivity in Banach spaces and the Mazur-Ulam theorem on isometries
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- by Julian Gevirtz PDF
- Trans. Amer. Math. Soc. 274 (1982), 307-318 Request permission
Abstract:
A mapping $f$ of an open subset $U$ of a Banach space $X$ into another Banach space $Y$ is said to be $(m,M)$-isometric if it is a local homeomorphism for which $M \geqslant {D^ + }f(x)$ and $m \leqslant {D^ - }f(x)$ for all $x \in U$, where ${D^ + }f(x)$ and ${D^ - }f(x)$ are, respectively, the upper and lower limits of $|f(y) - f(x)|/|y - x|\;{\text {as}}\;y \to x$. For $0 < \rho \leqslant 1$ we find a number $\mu (\rho ) > 1$ which has the following property: Let $X$ and $Y$ be Banach spaces and let $U$ be an open convex subset of $X$ containing a ball of radius $r$ and contained in the concentric ball of radius $R$. Then all $(m,M)$-isometric mappings of $U$ into $Y$ are injective if $M/m < \mu (r/R)$. We also derive similar injectivity criteria for a more general class of connected open sets $U$. The basic tool used is an approximate version of the Mazur-Ulam theorem on the linearity of distance preserving transformations between normed linear spaces.References
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Additional Information
- © Copyright 1982 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 274 (1982), 307-318
- MSC: Primary 46B20
- DOI: https://doi.org/10.1090/S0002-9947-1982-0670934-5
- MathSciNet review: 670934