How do you compute the dual norm of an induced norm on a subspace of a finite-dimensional $\ell^p$-normed vector space?

Question

Say you have a finite-dimensional vector space $V$ with an $\ell^p$ norm on it. In general, the norm induced on a subspace $V_s$ of doesn't have to be another $\ell^p$ norm, so the unit sphere in $V_s$ using this induced can be some strange shape.

Given $V_s$ and a norm $||·||$ induced this way on it, how can one compute an expression for the dual norm $||·||_*$ on $V_s^*$, the dual space of linear functionals on $V_s$?

I understand that this norm must satisfy the relationship $||w||_* = \text{sup }\frac{w(v)}{||v||}$ for $v$ in $V_s$ and $w$ in $V_s^*$, and that this means I need to find the intersection of the unit sphere in $V_s$ with the direction specified by $w$. However, I'm not sure what a good strategy might be to actually find an expression for the dual norm in this way. I thought that some implication of Hahn-Banach might help to pave the way forward, but after some research I still haven't seen anything obvious.

I do have a hunch that for the case where the norm on $V$ is $\ell^1$ or $\ell^\infty$, and hence where the unit sphere for induced norm on $V_s$ is some sort of polytope, that the unit sphere on $V_s^*$ will be the dual polytope exchanging faces and vertices.

Answer 1

An exact solution can be found here using the Hahn-Banach Theorem: http://math.unl.edu/~s-bbockel1/928/node25.htm

Using this, you can show that $V^∗_S$ is isometrically isomorphic to $V^∗/S$°, where $S°$ is the subspace in V∗ for which $s(t)=0$ for $s$ in $S°$ and $t$ in $S$. – Mike Battaglia Jul 21 at 4:06

Answer 2

This is a fairly general case of the Legendre transformation, I guess. I don't really see that it should be that much simpler than the general case (but I'm not an expert).