Proposition 5

Let $X$ be a compact space, $Y$ a normed space over $\mathbf{R}$ and $H$ a subset of $\mathscr{C}(X; \mathbf{R})$. If every continuous real-valued function on $X$ can be uniformly approximated by functions of $H$, then every continuous mapping $f$ of $X$ into $Y$ can be uniformly approximated by linear combinations of functions of $H$ with coefficients in $Y$.

Given any real number $\varepsilon > 0$, for each $x \in X$ there exists an open neighbourhood of $x$ in which the oscillation of $f$ is $\leq \varepsilon$. Hence there is a finite open covering $(A_i)_{1 \leq i \leq n}$ of $X$ such that the oscillation of $f$ in each $A_i$ is $\leq \varepsilon$. Let $a_i$ be a value of $f$ in $A_i$ ($1 \leq i \leq n$), and let $(u_i)_{1 \leq i \leq n}$ be a continuous partition of unity subordinate to the covering $(A_i)$ (Chapter IX, § 4, no. 4, Corollary to Proposition 4). Let $x$ be any point of $X$. For each index $i$ such that $x \notin A_i$, we have $u_i(x) = 0$, and for each index $i$ such that $x \in A_i$ we have $\|f(x) - a_i\| \leq \varepsilon$; it follows that

On the other hand, by hypothesis there is a function $v_i \in H$ such that

for all $x \in X$ ($1 \leq i \leq n$); hence we have

and the proof is complete.

From Proposition 5 it follows that, to each of the propositions in which we have proved that a certain subset $H$ of $\mathscr{C}(X; \mathbf{R})$ is dense, there corresponds an analogous proposition for continuous mappings of $X$ into an arbitrary normed space $Y$. We shall write down explicitly only the proposition which corresponds in this way to Theorem 3. Given a set $H$ of real-valued functions on $X$, a polynomial in the functions of $H$, with coefficients in $Y$, is defined to be any linear combination, with coefficients in $Y$, of products of a finite (possibly empty) family of functions belonging to $H$. Then: