 Dual test-function norm

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.