Corollary 1

If $X$ is a compact subset of the space $\mathbf{C}^n$, then every continuous mapping $(z_1, z_2, \ldots, z_n) \to f(z_1, z_2, \ldots, z_n)$ of $X$ into a normed space $Y$ over the field $\mathbf{C}$ can be uniformly approximated by polynomials in the $z_k$ and $\overline{z}_k$ with coefficients in $Y$.

We shall see later that in general it is not possible to approximate $f$ uniformly by polynomials (with coefficients in $Y$) in the variables $z_k$ alone, even if $Y = \mathbf{C}$.