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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
DOI: https://doi.org/10.1090/S0002-9939-1995-1260178-5
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 $\left \| T \right \| = 1$ such that $\left \| {T \circ \varphi (f) - f} \right \| \leq 6\varepsilon$ 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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Keywords: <!– MATH $\varepsilon$ –> <IMG WIDTH="15" HEIGHT="18" ALIGN="BOTTOM" BORDER="0" SRC="images/img1.gif" ALT="$\varepsilon$">-isometry, Banach space, linear operator, isometry
Article copyright: © Copyright 1995 American Mathematical Society