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Proceedings of the American Mathematical Society

Published by the American Mathematical Society, the Proceedings of the American Mathematical Society (PROC) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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$\epsilon$-isometric embeddings
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by Songwei Qian PDF
Proc. Amer. Math. Soc. 123 (1995), 1797-1803 Request permission

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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Additional Information
  • © Copyright 1995 American Mathematical Society
  • 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