Lemma 4

Let A be a commutative unital Banach algebra. Let $\Lambda$ be a set and $x= (x_{\lambda})_{\lambda\in\Lambda}$ a family of elements of A. If the unital subalgebra generated by the family $x$ is dense in A, then the simultaneous spectrum Sp$^{\Lambda}_A(x)$ is polynomially convex.

This follows from assertion b) of Prop. 12 of I, p. 43 and Definition 3.

Any intersection of polynomially convex subsets of $\mathbf{C}^{\Lambda}$ is polynomially convex (Lemma 3). This justifies the following definition: