Exercise 21, § 3

a) Let $(X_{\lambda})_{\lambda\in\Lambda}$ be a family of indeterminates. For every formal series $f\in \mathbf{C}[[(X_{\lambda})]]_{\lambda\in\Lambda}$, let $\|f\|$ be the sum of the absolute values of its coefficients. Let A be the set of the $f\in \mathbf{C}[[(X_{\lambda})]]_{\lambda\in\Lambda}$ such that $\|f\|<+\infty$. Equipped with the usual addition and multiplication, A is a commutative unital Banach algebra without radical.

b) For every commutative unital Banach algebra B, there exists an algebra A of the type of a) and a continuous unital morphism of A onto B.