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

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 specific algorithm for finding 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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There is no algorithm to find a closed form expression for the dual norm, even in two dimensions. For example, let the unit disk be $\{(x,y): x^2+|y|+p(x^2) \le 1 \}$ where $p$ is a polynomial of high degree with sufficiently small coefficients so that the unit disk is convex. Finding the dual norm of vector $(t, 1)$ ($t$ small) leads to the search for point on the graph of $y=1-x^2-p(x^2)$ with slope $-t$. This amounts* to solving $2x+2xp'(x^2)=t$. Since $p$ can be a generic polynomial of high degree (as long as the coefficients are small), there is no closed form for the solution.
(*) One can imagine finding the minimum without equating the derivative to zero; but the methods for doing so are algorithmic and do not produce a closed-form expression for the minimum in terms of $t$.
This is a great point. Thanks for this. Do you know of any conditions on the norm which are sufficient to deem the dual norm to have a closed form solution? I'm stuck specifically trying to figure out the various dual norms of norms which can be obtained by taking quotient and subspace norms of $\ell_p$ norms (say that five times fast...) –  Mike Battaglia Oct 8 '13 at 21:31