Lemma 5

Let X be a compact topological space, F a finite subset of $\mathscr{C}(X)$ and $\varepsilon >0$ a real number.

There exist a finite subset $X_0\subset X$ and a linear mapping $u:\mathscr{C}(X_0)\rightarrow \mathscr{C}(X)$ of norm $\leqslant 1$ such that $\|u(f|X_0)-f\|\leqslant \varepsilon$ for every $f\in F$.

Since the set F is an equicontinuous subset of $\mathscr{C}$ (X), there exists a finite covering $(U_i)_{i\in I}$ of X such that, for every $i\in I$ and every $f\in F$, the diameter of $f(U_i)$ is $\leqslant \varepsilon$. For $i$ in I, choose a point $x_i$ of $U_i$, and denote by $X_0$ the finite set of the $x_i$ for $i\in I$.

Let $(\varphi_i)_{i\in I}$ be a continuous partition of unity subordinate to $(U_i)_{i\in I}$ (TG, IX, p. 47, th. 3). Define a linear mapping $u$ of $\mathscr{C}(X_0)$ into $\mathscr{C}(X)$ by putting

for every $g\in \mathscr{C}(X_0)$. The linear mapping $u$ has norm $\leqslant 1$ and satisfies $\|u(f|X_0)-f\|\leqslant \varepsilon$ for every $f$ in F, whence the assertion.