Use the normalized <inner product>
$$
\langle g,h\rangle=\mathbb E_{n\leq N}g(n)h(n)=\frac1N\sum_{n\leq N}g(n)h(n).
$$
The <test-function seminorm> generated by $\mathcal F$ and its <dual test-function norm> are
$$
\|g\|_{\mathcal F}=\sup_{f\in\mathcal F}|\langle g,f\rangle|,
\qquad
\|g\|_{\mathcal F}^{*}=\sup_{\|h\|_{\mathcal F}\leq1}|\langle g,h\rangle|.
$$
The first quantity may only be a <seminorm> if $\mathcal F$ does not separate all functions in $\mathcal H_N$; correspondingly, the second may be infinite outside the linear span detected by $\mathcal F$.

Solution

Past exam of the mathematics course of the University of Cambridge / 2022 / iii / Paper 117 / 4 / b / Solution

{title2=$\|\cdot\|_{\mathcal F}$}

For a family $\mathcal F$ of real-valued functions on a finite set and normalized <inner product> $\langle\cdot,\cdot\rangle$, the test-function seminorm is
$$
\|g\|_{\mathcal F}=\sup_{f\in\mathcal F}|\langle g,f\rangle|.
$$
It measures the largest correlation of $g$ with an allowed test. It is a <seminorm>, and becomes a <norm> when the tests separate points.

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