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

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 $\lambda = \|{\bf 1}\|_E$.