Corollary 2

Let E be a Banach space. Let K be a polynomially convex compact subset of $\mathbf{C}^n$. Let P be the set of germs in a neighbourhood of K of polynomial functions on $\mathbf{C}^n$ with values in E. Then every element of $\mathscr{O}(K; E)$ is the limit of a sequence of elements of P.

Let E be provided with the multiplication defined by $ab= 0$ for all $a$ and $b$ in E (example 1 of I, p. 17). It is a commutative Banach algebra. Let A be the commutative unital Banach algebra obtained from E by adjunction of a unit element. Since the canonical mapping $\mathscr{O}(K; A)\rightarrow \mathscr{O}(K; E)$ is continuous, the assertion results from the Oka-Weil theorem applied to the algebra A.

For $n= 1$ and $A =\mathbf{C}$, one also has the following result, which will be made more precise by corollary 2 of I, p. 150.