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 $(E,\Vert \cdot {\Vert }_E)$ 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 $\Vert \cdot {\Vert }_E=\lambda \Vert \cdot {\Vert }_{{Ł}_p}$, where $\lambda =\Vert 1{\Vert }_E$.