= Kolmogorov criterion for uniform approximation
{c}
For real $f$, a candidate $u\in U\subset C(K,\mathbb R)$, and active set $E=\{x:|f(x)-u(x)|=\|f-u\|_\infty\}$, the candidate is best precisely when every $v\in U$ has some $x\in E$ with $(f(x)-u(x))v(x)\leq0$. A direction with strictly matching error signs on the whole active set gives a uniformly improving small perturbation by compactness. Conversely one oppositely signed active value already prevents improvement. This criterion yields the <Chebyshev alternation theorem>.
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