= Lattice approximation from two-point approximation
Suppose a family $A\subseteq C(K,\mathbb R)$ on a <compact space> is closed under pointwise maximum and minimum. If a target $f$ can be approximated arbitrarily closely at each pair of points by a member of $A$, it can be approximated in the <supremum norm>. For a fixed first point, take finitely many pairwise approximants whose neighborhoods cover $K$ and take their maximum: it lies above $f-\varepsilon$ everywhere and below $f+\varepsilon$ near that first point. Take finitely many such first-point neighborhoods covering $K$ and take the minimum of their maxima. The result is within $\varepsilon$ of $f$ everywhere. Starting with tolerance $\varepsilon/2$ ensures a strict <uniform approximation> error smaller than $\varepsilon$.
Back to article page
