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

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, 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.

For a test-function seminorm on a finite set, define the dual functionalIf the tests span the whole function space, the original seminorm is a norm, and this is its finite dual norm. Without that spanning condition it can be infinite. Indeed, a component of $g$ orthogonal to the test span pairs nontrivially with a direction on which the original seminorm vanishes; arbitrary multiples of that direction remain admissible. The dual functional is finite precisely on the test span.

For a closed, convex, symmetric and uniformly bounded test family $\mathcal{F}$, the Bipolar theorem for a dual pair identifies the dual unit ball with $\mathcal{F}$, including the lower-dimensional case. Submultiplicativity under pointwise products then controls powers of a test in a polynomial, the step used by the dense model theorem for a multiplicative test family.