= Test-function seminorm
{title2=$\|\cdot\|_{\mathcal F}$}
Let $X$ be a nonempty finite <set>, and let $\mathcal F$ be a nonempty, uniformly bounded family of <real-valued functions> on $X$. With the normalized <inner product>, define
$$
\|g\|_{\mathcal F}=\sup_{f\in\mathcal F}|\langle g,f\rangle|.
$$
Uniform boundedness makes the supremum finite. Absolute homogeneity and the <triangle inequality> follow from the same properties of each <linear functional>. This is a <norm> exactly when the <linear span> of $\mathcal F$ is all of $\mathbb R^X$, so no nonzero $g$ is <orthogonal> to every test.<Decompositions, approximate structure, transference, and the Hahn-Banach theorem by Timothy Gowers>[Section 3, Corollary 3.5]{ref}
Separating the underlying <points> of $X$ is insufficient. On two <points>, the single test $f=(1,2)$ takes distinct values, yet the nonzero <vector> $g=(2,-1)$ has zero test <seminorm>. The tests must detect differences between whole <functions>.
Back to article page
