# Closed-form expressions for dual norms of real normed vector spaces

Didn't get any biters over at MSE, so I figure this place might be more appropriate...

Say that $V$ is a finite-dimensional real normed vector space, where for some $v \in V$ the norm is notated by $\|v\|$. Then say that $V^*$ is the dual space of linear functionals on $V$.

The "dual norm" $\| \cdot \|^*$ naturally exists on $V^*$, given for some $v^* \in V^*$ by

$\displaystyle\|v^*\|^* = \sup_{v \in V} \left\{ \frac{|\langle v^*, v\rangle|}{\|v\|} \right\} = \sup_{v \in V} \left\{ |\langle v^*, v\rangle| : \|v\| = 1 \right\}$

for $\langle v^*,v \rangle$ the dual pairing between $v^*$ and $v$.

If a basis is chosen for $V$, then is vectors can be represented as $\mathbb{R}$-tuples. Sometimes the original norm on $V$ can be denoted by a closed-form expression on the entries of this tuple. For instance, the $\ell_p$ norm on a vector $(x_1, x_2, ..., x_n)$ is given by $\left(|x_1|^p + |x_2|^p + ... + |x_n|^p\right)^{\frac{1}{p}}$.

In these cases, is there a general algorithm to find a closed-form expression for the dual norm given the dual basis of $V^*$?

(Obviously the solution is well known for the $\ell_p$ norm, but what about in general?)

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For a general question like this, the general answer is: You find the dual norm by any method which allows you to solve the respective maximization problem. I don't think that one can say something more specific. – Dirk Sep 28 '13 at 13:22
There need not always be closed form expressions; there can also be examples where computing the dual norm is NP-Hard etc.; this question is not really precise or clear in what it is asking... – Suvrit Sep 28 '13 at 17:34
Dirk: but is there any specific algorithm that works in general to handle the problem for finite-dimensional vector spaces? – Mike Battaglia Sep 28 '13 at 19:44
suvrit: I'm looking for a specific algorithm to symbolically compute an expression for the dual norm (when possible). What's an example of a finite-dimensional normed real vector space in which finding the dual norm is NP-hard? – Mike Battaglia Sep 28 '13 at 19:45
Edited the question to make it clearer that I'm looking for some specific method or algorithm here. – Mike Battaglia Sep 28 '13 at 19:45