Isometry/Set distances/Exercise

Let $V$ and $W$ denote Euclidean vector spaces, and let

$\phi :V⟶W$

be a linear mapping. Show that $\phi$ is an isometry if and only if for arbitrary subsets $A,B\subseteq V$ the equation

$d(\phi (A),\phi (B))=d(A,B)$

holds.

Create a solution