Let $K$ be a compact [Hausdorff space](/page/Hausdorff%20Space), let $C(K)$ denote the real [Banach space](/page/Banach%20Space) of continuous functions $K \to \mathbb{R}$ with the uniform norm $\|\cdot\|_\infty$, let $X \subset C(K)$ be a finite-dimensional linear subspace, let $f \in C(K)$, and let $p \in X$. Define $e := f - p$ and $E := \|e\|_\infty$.

If $E = 0$, then $p$ is a best uniform approximation to $f$ from $X$.

If $E > 0$, define the active set

\begin{align*} A := \{t \in K : |e(t)| = E\}. \end{align*}

Then $p$ is a best uniform approximation to $f$ from $X$ if and only if

\begin{align*} 0 \in \operatorname{conv}\{\operatorname{sgn}(e(t))\,\delta_t|_X : t \in A\} \subset X^*, \end{align*}

where $\delta_t|_X: X \to \mathbb{R}$ is the restricted evaluation functional $x \mapsto x(t)$.