 Test-function seminorm

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, defineUniform 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.^[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.