Corollary 2

Let $X$ be a locally compact space and let $C_0(X)$ be the normed $\mathbf{C}$-algebra of continuous mappings of $X$ into $\mathbf{C}$ which tend to 0 at infinity. Let $A$ be a subalgebra of $C_0(X)$ which separates the points of $X$ and is such that (i) $\overline{f} \in A$ whenever $f \in A$, (ii) for each $x \in X$, there is an $f \in A$ such that $f(x) \neq 0$. Then $A$ is dense in $C_0(X)$.

If $X'$ is the compact space obtained by adjoining a point at infinity $\omega$ to $X$, then $C_0(X)$ can be identified with the subspace of $C(X; \mathbf{C})$ consisting of continuous mappings which vanish at $\omega$, the norm on $C_0(X)$ being defined by

By virtue of Proposition 7, every $f \in C_0(X)$ can be uniformly approximated by polynomials with complex coefficients in the functions belonging to $A$; moreover, since $f(\omega) = 0$, the argument of no. 2, Proposition 4 shows that we may suppose these polynomials to have no constant term, and then they belong to $A$.

Another application of Proposition 7 is the following: