My textbook says:
Frobenius norm defined on $\mathbb{R}^{m,n}$ by a formula: $$\|A\|_F=\sqrt{\sum_{i=1}^n\sum_{j=1}^m|a_{i,j}|^2}$$
when $n>1, \ m>1$ is not induced by any vector's $p$-norm.
But there is no proof. I searched over the Internet and didn't find one. Is it very hard to prove? I would be very grateful for help.
Best Answer
Any induced norm of the identity matrix is $1.$