Test-function seminorm ($\Vert \cdot {\Vert }_{\mathcal{F}}$)

For a family $\mathcal{F}$ of real-valued functions on a finite set and normalized inner product $⟨\cdot ,\cdot ⟩$, the test-function seminorm isIt measures the largest correlation of $g$ with an allowed test. It is a seminorm, and becomes a norm when the tests separate points.

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.