Proposition 7

Let $X$ be a compact space and let $H$ be a set of continuous complex-valued functions on $X$ which separates the points of $X$. Then every continuous mapping of $X$ into a normed space $Y$ over $\mathbf{C}$ can be uniformly approximated by polynomials in the functions $f \in H$ and their conjugates $\overline{f}$ with coefficients in $Y$.

We have only to note that $Y$ is also a normed space over $\mathbf{R}$ and to apply Proposition 6 to the set of real parts and imaginary parts of the functions $f \in H$, using the formulas