# Is the dual norm of an induced norm an induced norm?

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.

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 $$\|I_n\|_{\operatorname{tr}}=n\ne1$$.