Exercise 2, § 2

Soit $n\geqslant 0$ an integer. Let $A_n$ be the complex commutative Banach algebra of functions $f: [0,1]\rightarrow \mathbf{C}$ admitting continuous derivatives in $[0,1]$ up to order $n$, endowed with the norm

$n$

$\|f\|=\sum k1$! sup$_{0\leqslant t\leqslant 1}|f^{(k)}(t)|$

$k=0$

(example 4 of I, p. 18). Now prove that $A_n$ is isomorphic to $A_m$ if, and only if, $n=m$. (However, according to example 1 of I, p. 144, these Banach algebras all have the same character space).