Exercise 11, § 3

Let B be the algebra over $\mathbf{C}$ of indefinitely differentiable functions from $[0,1]$ into $\mathbf{C}$.

(a) Let A be a subalgebra of B, provided with a norm making A a Banach algebra. Now prove that there exists a sequence $(m_n)_{n\geqslant 0}$ in $\mathbf{R}_+$ such that, for every $x\in A$, one has

sup$_{0\leqslant t\leqslant 1}|x^{(n)}(t)|= O(m_n)$.

(Let $B_n$ be the Banach algebra of functions $n$ times continuously differentiable on $[0,1]$. According to prop. 9 of I, p. 40, the canonical injection $A\rightarrow B_n$ is continuous; let $M_n$ be its norm; if $x\in A$, one has sup$_{0\leqslant t\leqslant 1}|x^{(n)}(t)|\leqslant n$!M$_n\|x\|.$)

(b) There exists no norm on B which makes it a Banach algebra.