Theorem 2 (Oka–Weil)

Let K be a polynomially convex compact subset of $\mathbf{C}^n$ and P the set of germs in a neighbourhood of K of polynomial functions on $\mathbf{C}^n$ with coefficients in A. Then P is dense in $\mathscr{O}(K; A)$. More precisely, every element of $\mathscr{O}(K; A)$ is the limit of a sequence of elements of P.

Consider an element of $\mathscr{O}(K; A)$, germ of a function $f\in$ $\mathscr{O}(U; A)$, where U is an open neighbourhood of K. By lemma 7 of I, p. 48, there exists a compact neighbourhood L of K contained in U which is polynomially convex. Let V be the interior of L; it is a neighbourhood of K.

By the preceding proposition, there exists a sequence $(P_k)$ of polynomial functions on $\mathbf{C}^n$ with coefficients in A which converges to $f|L$ in $\mathscr{C}(L; A)$. In particular, the sequence $(P_k)$ converges to $f|V$ in $\mathscr{O}(V; A)$.

By definition of the topology on $\mathscr{O}(K; A)$ (cf. EVT, II, p. 29, prop. 5), the canonical mapping of $\mathscr{O}(V; A)$ into $\mathscr{O}(K; A)$ is continuous. Consequently, the sequence of germs in a neighbourhood of K of the functions $P_k$ converges to the germ of $f$ in a neighbourhood of K in the space $\mathscr{O}(K; A)$, which was to be proved.