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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Injectivity in Banach spaces and the Mazur-Ulam theorem on isometries

Author: Julian Gevirtz
Journal: Trans. Amer. Math. Soc. 274 (1982), 307-318
MSC: Primary 46B20
MathSciNet review: 670934
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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 $ \vert f(y) - f(x)\vert/\vert y - x\vert\;{\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.

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Keywords: Quasi-isometric mapping, injectivity, uniform domain
Article copyright: © Copyright 1982 American Mathematical Society