Question about vals and induced Lp norms

Suppose you have the space of vals, and you impose a linear
transformation L on this space sending vals to G-svals. Dual to this
new space of svals is the space of smonzos, and there's a natural dual
transformation to L which sends smonzos back to monzos. We can then
use the Lp norm on monzos to induce a norm on smonzos, and then use
the dual norm of these smonzos to induce a norm on svals. I'll call
this norm which is dual to the induced norm G-Lp*.

Note that the preimage of any sval in L is going to be an affine
subspace of vals. I then have the following question: given any sval
and the G-Lp* norm, will the norm of the sval be the same as the
minimum of the Lp* norms of the set of vals in its preimage?

-Mike