= Schauder approximation lemma
{c}
{title2=$\sup_{x\in K}\|T(x)-T_\varepsilon(x)\|\leq\varepsilon$}
Let $K$ be a nonempty compact <convex set> in a <Banach space> and $T:K\to K$ a <continuous map>. For every $\varepsilon>0$, there is a finite-dimensional <linear subspace> $L_\varepsilon$ and a <continuous map> $T_\varepsilon:K\to K\cap L_\varepsilon$ that approximates $T$ uniformly to within $\varepsilon$ in <norm>.
Choose $y_1,\ldots,y_N\in T(K)$ whose open radius-$\varepsilon$ <balls> cover the compact image $T(K)$. Set $w_i(x)=\max\{0,\varepsilon-\|Tx-y_i\|\}$ and
$$
T_\varepsilon(x)=\frac{\sum_iw_i(x)y_i}{\sum_iw_i(x)}.
$$
The denominator is positive. This <convex combination> belongs to $K\cap\operatorname{span}\{y_1,\ldots,y_N\}$ and depends continuously on $x$. Every contributing $y_i$ has <distance> less than $\varepsilon$ from $Tx$, so the <triangle inequality> gives $\|T_\varepsilon(x)-Tx\|<\varepsilon$.
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