Induced Norm

Let $V$ and $W$ be normed vector spaces with norms $p$ and $q$ respectively. Let $A:V \mapsto W$ and let $\pmb x \in V$ .

The definition of the induced norm on $A$ is

\begin{align*} \sup_{\pmb x} \frac{\lVert A \pmb x \rVert_q}{\lVert \pmb x \rVert_p} \end{align*}