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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

$ \epsilon$-isometric embeddings


Author: Songwei Qian
Journal: Proc. Amer. Math. Soc. 123 (1995), 1797-1803
MSC: Primary 46B04; Secondary 46E30
MathSciNet review: 1260178
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Abstract: In this paper we study into $ \varepsilon $-isometries. We prove that if $ \varphi $ is an $ \varepsilon $-isometry from $ {L^p}({\Omega _1},{\Sigma _1},{\mu _1})$ into $ {L^p}({\Omega _2},{\Sigma _2},{\mu _2})$ (for some $ p, 1 < p < \infty $ ), then there is a linear operator $ T:{L^p}({\Omega _2},{\Sigma _2},{\mu _2}) \mapsto {L^p}({\Omega _1},{\sigma _1},{\mu _1})$ with $ \left\Vert T \right\Vert = 1$ such that $ \left\Vert {T \circ \varphi (f) - f} \right\Vert \leq 6\varepsilon $ for each $ f \in {L^p}({\Omega _1},{\Sigma _1},{\mu _1})$. This forms a link between an into isometry result of Figiel and a surjective $ \varepsilon $-isometry result of Gevirtz in the case of $ {L^p}$ spaces.


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DOI: http://dx.doi.org/10.1090/S0002-9939-1995-1260178-5
PII: S 0002-9939(1995)1260178-5
Keywords: $ \varepsilon $-isometry, Banach space, linear operator, isometry
Article copyright: © Copyright 1995 American Mathematical Society