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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.

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

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  • $\begingroup$ Correct! In addition: $||I||_F = \sqrt{\min\{m,n\}}$ hence $||I||_F > 1$ for $n>1, m>1$. $\endgroup$ – TheWaveLad Dec 1 '13 at 19:54
  • $\begingroup$ brilliant! thank you very much $\endgroup$ – xan Dec 1 '13 at 20:10

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