- $\|A\|_p = \displaystyle \max_{\|x\|_p = 1} \|Ax\|_p $
- $\|A\|_2 \leq \|A\|_F \leq \sqrt{n}\|A\|_2$
How I can show that $1$ and $2$ are correct?
$2)$ $||Ax||_{2}=\sqrt{\sum_{i=1}^{n} | \sum_{j=1}^{n} a_{kj},x_{j}|^{2}}\leq \sqrt{[\sum_{k=1}^{n}(\sum_{j=1}^{n} |a_{kj}|^2)(\sum_{j=1}|x_{j}|^{2})]} = ||A||_{F} ||x||_{2} $
but how i show $||A||_{F}\leq \sqrt{n}||A||_{2}$
$||A||_{F} \leq (\sqrt{n*\sum_{i=1}^{n} | \sum_{j=1}^{n} a_{kj},x_{j}|^{2}})$
with $n>1$