Consider
$x = A\,b$,
where $x \in \mathbb R^{n\times1}$, $b \in \mathbb R^{n\times1}$ and $A \in \mathbb R^{n\times n}$.
I'am looking for a bound of $x$ as
$\|x\| \leq \|Ab\|$
Is it correct to write the following?
$\|x\| \leq \|A\|_i\|b\|$, where $\|\cdot\|$ is the Euclidean norm and $\|\cdot\|_i$ is an induced norm for matrices.
I'am not sure if Cauchy-Schwarz Inequality can be used here.
The operator norm is defined this way $$ \|A\|^* = \sup_{\|x\|=1} \|Ax\|. $$ This definition is in such a way that $\|A\|^*<\infty$ if and only if $A$ is continuous (we are talking about linear maps). Indeed $$ \|Ab\| = \|b\| \left\| A \frac{b}{\|b\|} \right\| \leq \|b\| \sup_{b\neq 0} \left\| A \frac{b}{\|b\|} \right\| = \|b\|\|A\|^*. $$