= Dual test-function norm
{title2=$\|\cdot\|_{\mathcal F}^{*}$}
For a <test-function seminorm> on a finite <set>, define the dual functional
$$
\|g\|_{\mathcal F}^*=\sup_{\|h\|_{\mathcal F}\leq1}|\langle g,h\rangle|.
$$
If 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>.
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