Frobenius norm is not induced

normed-spaces

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.


Any induced norm of the identity matrix is $1.$

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