Dual test-function norm
= Dual test-function norm
{title2=$\|\cdot\|_{\mathcal F}^{*}$}
The dual test-function norm is
$$
\|g\|_{\mathcal F}^{*}=\sup_{\|h\|_{\mathcal F}\leq1}|\langle g,h\rangle|.
$$
If $\mathcal F$ is closed, convex, and symmetric, the <Bipolar theorem for a dual pair> identifies its dual unit ball with $\mathcal F$. Submultiplicativity under pointwise products allows a polynomial in one test function to remain controlled in this norm.