Dual test-function norm ($\Vert \cdot {\Vert }_{\mathcal{F}}^{\ast }$)

The dual test-function norm isIf $\mathcal{F}$ 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.