Exercise 5, § 8

Let $A$ be the Banach algebra of functions from $[0,1]$ into $\mathbf{C}$ admitting continuous derivatives in $[0,1]$ up to order $n$ (example 4 of I, p. 18). Let $\omega \in [0,1] =\mathsf{X}(A)$.

(a) The smallest closed ideal I of A such that $V(I) =\{\omega \}$ is the set of $f\in A$ such that $f, f', . . . , f^{(n)}$ vanish at $\omega$. (Use prop. 5 of I, p. 94) (b) If $g\in A$, the norm of its canonical image in $A/I$ is equal to

$\sum^n|g^(\overset{k)}{k}($!$\omega )|$.

$k=0$

(c) The algebra $A/I$ is isomorphic to $\mathbf{C}[X]/(X^{n+1})$; the only maximal ideal of A containing I is the set of $f\in A$ such that $f(\omega ) = 0$.