Exercise 29, § 3

*Let X be a compact real manifold. We equip $\mathscr{C}^{\infty}(X)$ with the topology of uniform convergence of the functions and of all their derivatives; it is a Fréchet space. Let A be a subalgebra of $\mathscr{C}(X)$. Suppose that A is provided with a norm $f\mapsto \|f\|_A$ which makes it a Banach algebra, such that the canonical injection of A into $\mathscr{C}(X)$ is continuous. Suppose moreover that $\mathscr{C}^{\infty}(X)$ is contained and dense in A.

a) There exists a constant $C\geqslant 0$ and an integer $k\in \mathbf{N}$ such that for all $f\in \mathscr{C}^{\infty}(X)$, one has

$\|f\|_A\leqslant C$ sup$_{x\in X}|f^{(k)}(x)|$.

b) The subalgebra A is full in $\mathscr{C}(X)$. (Let $f\in A$ not vanishing on X and $\delta =$ inf$_{x\in X}|f(x)|$; choose $g\in \mathscr{C}^{\infty}(X)$ such that $\|f-g\|_A< \delta /3$, then note that $f^{-1}=g^{-1}\sum_{n\in\mathbf{N}}((g-f)/g)^n$ in $\mathscr{C}(X)$, and verify that the series converges in A.)

c) Deduce a new proof of the Wiener theorem (I, p. 38, example).*