Let $R$ form a D273: Division ring.
Let $N$ and $M$ each form a D4718: Norm-metrised vector space over $R$.
Let $\Vert \cdot \Vert_N$ and $\Vert \cdot \Vert_M$ each be the D306: Vector space norm in $N$ and $M$, respectively.
Let $f : N \to M$ be a D705: Proportionally bounded linear map with respect to $N$ and $M$.
Let $N$ and $M$ each form a D4718: Norm-metrised vector space over $R$.
Let $\Vert \cdot \Vert_N$ and $\Vert \cdot \Vert_M$ each be the D306: Vector space norm in $N$ and $M$, respectively.
Let $f : N \to M$ be a D705: Proportionally bounded linear map with respect to $N$ and $M$.
Then $f$ is a D47: Lipschitz map with respect to $N$ and $M$.