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
DOI:
https://doi.org/10.1090/S0002-9947-1982-0670934-5
MathSciNet review:
670934
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Abstract | References | Similar Articles | Additional Information
Abstract: A mapping of an open subset
of a Banach space
into another Banach space
is said to be
-isometric if it is a local homeomorphism for which
and
for all
, where
and
are, respectively, the upper and lower limits of
. For
we find a number
which has the following property: Let
and
be Banach spaces and let
be an open convex subset of
containing a ball of radius
and contained in the concentric ball of radius
. Then all
-isometric mappings of
into
are injective if
. We also derive similar injectivity criteria for a more general class of connected open sets
. 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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Additional Information
DOI:
https://doi.org/10.1090/S0002-9947-1982-0670934-5
Keywords:
Quasi-isometric mapping,
injectivity,
uniform domain
Article copyright:
© Copyright 1982
American Mathematical Society