\begin{definition}[Definition:Equivalence of Norms]
Let $\norm {\,\cdot\,}_1$ and $\norm {\,\cdot\,}_2$ be norms on a vector space $V$.
$\norm {\,\cdot\,}_1$ and $\norm {\,\cdot\,}_2$ are '''equivalent''' {{iff}} there exist real constants $c$ and $C$ such that:
:$\forall \mathbf x \in V: c \norm {\mathbf x}_1 \le \norm {\mathbf x}_2 \le C \norm {\mathbf x}_1$
\end{definition}