Let $T$ be a linear operator between finite-dimensional complex vector spaces $A, B$. Then it is mentioned on Wikipedia that for any $(p, q)$ there is a relation between the operator norms
$$||T : \ell_p A \to \ell_q B ||_{op} = ||T^* : \ell_{q'} B \to \ell_{p'} A ||_{op}$$
where $(p',q')$ are Holder conjugate to $(p,q)$. What is the quickest way to see this?
I know that by definition of the adjoint and Cauchy-Schwarz we can establish the case where $(p,q) = (p',q') = (2,2)$, and I imagine the proof I seek uses the more general Hölder's inequality. But since we're not dealing with inner product spaces generally, I don't know what relevant details of the adjoint to invoke.