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 $Ł_p [0, 1]$ for some $1\le p<\infty$, then $E$ coincides with $Ł_p [0, 1]$ and furthermore $\|\cdot\|_E = λ\|\cdot\|_{Ł_p}$, where $λ= \|{\bf 1}\|_E$.