Test-function seminorm
= Test-function seminorm
{title2=$\|\cdot\|_{\mathcal F}$}
For a family $\mathcal F$ of real-valued functions on a finite set and normalized <inner product> $\langle\cdot,\cdot\rangle$, the test-function seminorm is
$$
\|g\|_{\mathcal F}=\sup_{f\in\mathcal F}|\langle g,f\rangle|.
$$
It measures the largest correlation of $g$ with an allowed test. It is a <seminorm>, and becomes a <norm> when the tests separate points.