A rearrangement invariant space isometric to $L_p$ coincides with $L_p$

Yuri A. Abramovich; Mikhail Zaidenberg

A rearrangement invariant space isometric to $L_p$ coincides with $L_p$

Functional Analysis 2009-09-25 v1

Authors: Yuri A. Abramovich , Mikhail Zaidenberg

Abstract

The following theorem is the main result of this note. Theorem 1. Let $(E, \|\cdot\|_E)$ be a rearrangement invariant Banach function space on the interval $[0, 1]$ . If $E$ is isometric to $\L_p [0, 1]$ for some $1\le p<\infty$ , then $E$ coincides with $\L_p [0, 1]$ and furthermore $\|\cdot\|_E = \lambda\|\cdot\|_{\L_p}$ , where $\lambda = \|{\bf 1}\|_E$ .

Keywords

rearrangement invariant space banach function space isometric l_p space theorem

Cite

@article{arxiv.math/9509213,
  title  = {A rearrangement invariant space isometric to $L_p$ coincides with $L_p$},
  author = {Yuri A. Abramovich and Mikhail Zaidenberg},
  journal= {arXiv preprint arXiv:math/9509213},
  year   = {2009}
}

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