# Continuous linear operator norm

My book says that $||Ax|| \leq M ||x|| \ \forall x \in X \implies ||A|| \leq M$, because $||A|| = sup_{||x|| \leq 1} \ ||Ax|| \in [0, M]$.

However, I'm unable to see how this follows from the definition. Thank you very much.

• Because $\sup_{\|x\| \le 1} \|Ax\| \le M \sup_{\|x\| \le 1} \|x\|$.... – user296602 May 10 '16 at 20:01

For any $x$ with $\|x\|=1$, $$\|Ax\|\leq M.$$ So $M$ is an upper bound for the set $$\{\|Ax\|:\ \|x\|\leq1\}.$$ And thus $$\sup\{\|Ax\|:\ \|x\|\leq1\}\leq M.$$