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 $\|x+\alpha y\|\geq \|x\|$ for every $\alpha \in R$. We prove that every operator from $E$ into itself preserving orthogonality is an isometry multiplied by a constant.