= Solution
The <Riesz-Markov-Kakutani representation theorem> identifies the continuous dual of $C(\mathbb T)$ with the finite complex <Radon measures> on $\mathbb T$: every bounded complex-<linear functional> $L$ has a unique representation
$$
\boxed{L(f)=\int_{\mathbb T}f\,d\mu,\qquad \|L\|=|\mu|(\mathbb T).}
$$
The right side is the total variation norm of the <measure>. On this compact <metric space> <Radon measures> are regular Borel <measures>. A functional is positive precisely when its representing <measure> is positive; then $\|L\|=L(1)=\mu(\mathbb T)$.
Back to article page
