In spaces $V$ and $W$, define vector norms $\| \cdot \|_V $ and $\| \cdot \|_W $, respectively. Consider the operator norm induced by norms $\| \cdot \|_V $ and $ \| \cdot \|_W $
$$ \| A \| = \sup{\frac{\|Ax\|_W}{\|x\|_V}} $$
Do there exist norms $ \|\cdot\|_V $ and $ \|\cdot\|_W $ such that the Frobenius norm is induced by these norms?
It is well known that for any matrix norm induced by $ \|\cdot\| $, $ \|I\| = 1 $. But it is not truth for two different norms in $V$ and $W$. Can anybody help?
Since the argument on the norm of the identity does not work here, let's try the unitary invariance. The Frobenius norm is unitarily invariant ($\|A\|_F=\|PAQ^*\|_F$ for any unitary $P$ and $Q$ of suitable dimensions), so if there were such vector norms inducing the Frobenius norm, they would need to be unitarily invariant as well. The only unitarily invariant vector norm is the Euclidean norm (or its positive multiple), which induces the spectral norm.