Subspaces of Lp Isometric to Subspaces of ℓ p

Subspaces of Lp Isometric to Subspaces of ℓ p We present three results on isometric embeddings of a (closed, linear) subspace X of Lp=Lp[0,1] into ℓ p . First we show that if p ∉ 2N, then X is isometrically isomorphic to a subspace of ℓ p if and only if some, equivalently every, subspace of Lp which contains the constant functions and which is isometrically isomorphic to X, consists of functions having discrete distribution. In contrast, if p ∈ 2N; and X is finite-dimensional, then X is isometrically isomorphic to a subspace of ℓ p , where the positive integer N depends on the dimension of X, on p , and on the chosen scalar field. The third result, stated in local terms, shows in particular that if p is not an even integer, then no finite-dimensional Banach space can be isometrically universal for the 2-dimensional subspaces of Lp . http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Positivity Springer Journals

Subspaces of Lp Isometric to Subspaces of ℓ p

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Publisher
Kluwer Academic Publishers
Copyright
Copyright © 1998 by Kluwer Academic Publishers
Subject
Mathematics; Fourier Analysis; Operator Theory; Potential Theory; Calculus of Variations and Optimal Control; Optimization; Econometrics
ISSN
1385-1292
eISSN
1572-9281
D.O.I.
10.1023/A:1009764511096
Publisher site
See Article on Publisher Site

Abstract

We present three results on isometric embeddings of a (closed, linear) subspace X of Lp=Lp[0,1] into ℓ p . First we show that if p ∉ 2N, then X is isometrically isomorphic to a subspace of ℓ p if and only if some, equivalently every, subspace of Lp which contains the constant functions and which is isometrically isomorphic to X, consists of functions having discrete distribution. In contrast, if p ∈ 2N; and X is finite-dimensional, then X is isometrically isomorphic to a subspace of ℓ p , where the positive integer N depends on the dimension of X, on p , and on the chosen scalar field. The third result, stated in local terms, shows in particular that if p is not an even integer, then no finite-dimensional Banach space can be isometrically universal for the 2-dimensional subspaces of Lp .

Journal

PositivitySpringer Journals

Published: Oct 14, 2004

References

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