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I think this Q&A gets close to answering this, but does not provide a full response.
If you have some induced matrix norm $\|\cdot\|_{\|\cdot\|', \|\cdot\|'}$ induced by the vector norm $\|\cdot\|'$, I'm wondering if is it the dual of this norm (i.e. $\|\cdot\|_{\|\cdot\|', \|\cdot\|'}^D$) also an induced norm?
Furthermore, if $\|\cdot\|_{\|\cdot\|', \|\cdot\|'}^D$ is an induced norm, is there a relationship between the vector norm $\|\cdot\|'$ and the vector norm which would induce $\|\cdot\|_{\|\cdot\|', \|\cdot\|'}^D$?

I was considering trying to work with what I think is just an unnamed Theorem, $$ \|A\|_{\|\cdot\|, \|\cdot\|} = \|A^*\|_{\|\cdot\|^D, \|\cdot\|^D}, $$ but that doesn't seem to be getting me anywhere.

Daniele Tampieri
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Evan Mata
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1 Answers1

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No. E.g. the dual norm of the induced $2$-norm is the trace norm (a.k.a. nuclear norm or Schatten $1$-norm), but for $2\times2$ or larger matrices, the trace norm is not an induced norm because $\|I_n\|_{\operatorname{tr}}=n\ne1$.

user1551
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