Operators preserving orthogonality are isometries

Abstract

Let $E$ be a real Banach space. For $x,y\in E,$ we follow R.James in saying that $x$ is orthogonal to $y$ if $\Vert x+\alpha y\Vert \ge \Vert x\Vert$ for every $\alpha \in R$. We prove that every operator from $E$ into itself preserving orthogonality is an isometry multiplied by a constant.

Cited by in corpus (1)

• On T-orthogonality in Banach spaces