What is mean by a matrix norm induced by the following

Question

What does it mean to have the following situation:

A matrix norm is induced by

$A : (\mathbb{R}^n,|| \cdot ||_{1}) \rightarrow (\mathbb{R}^n,|| \cdot ||_{\infty})$

Are you taking the 1 norm of the matrix, then just simply applying the infinity matrix norm to that result?

Answer 1

The induced norm $(\mathbb R^n, \|\cdot\|_1) \to (\mathbb R^n, \|\cdot\|_{\infty})$ is defined by $$ \|A\| := \sup\limits_{\|x\|_1 \leq 1} \|Ax\|_{\infty} $$