= Compact Korovkin test space
Let $K$ be a <compact Hausdorff space>. A test subspace $H\subset C(K,\mathbb R)$ containing a strictly positive function is sufficient for <uniform convergence> of <positive linear operators on continuous functions> if, for each $x\in K$, the only <positive linear functional> agreeing with point evaluation on $H$ is that evaluation itself. By the <Riesz-Markov-Kakutani representation theorem>, this is uniqueness of the positive measure with the prescribed moments. Boundedness from the positive test function and the <Banach-Alaoglu theorem> prove the assertion by compactness. When $1\in H$, nonnegative tests with a unique zero at each prescribed $x$ give a convenient sufficient condition.
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