𝜀-isometric embeddings

Qian, Songwei

doi:10.1090/S0002-9939-1995-1260178-5

$\epsilon$-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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Sur les transformations isométriques d’espace vectoriels normés

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