Definition:Equivalence of Norms

Definition
Let $\norm {\,\cdot\,}_1$ and $\norm {\,\cdot\,}_2$ be norms on a vector space $V$.

$\norm {\,\cdot\,}_1$ and $\norm {\,\cdot\,}_2$ are equivalent there exist real constants $c$ and $C$ such that:
 * $\forall \mathbf x \in V: c \norm {\mathbf x}_1 \le \norm {\mathbf x}_2 \le C \norm {\mathbf x}_1$