Exercise 25, § 3

Let A be a locally $m$-convex complete commutative unital algebra (cf. I, p. 165, exerc. 37).

a) An element $x$ of A is invertible if and only if $\chi (x)\not= 0$ for every continuous character $\chi$ of A. The radical of A is the intersection of the kernels of the continuous characters of A, hence is closed.

b) In example e) of Exercise 37 of I, p. 165, let I be the ideal formed by the $f$ such that $f(n) = 0$ for every sufficiently large integer $n >0$. Then every maximal ideal containing I is dense and of infinite codimension.