Exercise 5, § 8

a) Let V be a real vector space, $f$ a non-zero linear form on V, H the kernel of $f$. Then, a function $\varphi$ on V of class $C^{\infty}$ vanishes on H if and only if it can be written as $f \varphi '$, where $\varphi '$ is a function of class $C^{\infty}$ on V.

b) Let V be a real vector space, $(f_i)_{i\in I}$ a finite family of non-zero linear forms such that the $H_i=$ Ker $f_i$ are pairwise distinct. Then, a function $\varphi$ on V of class $C\prod^{\infty}$ vanishes on the union of the $H_i$ if and only if it can be written as $\psi_{i\in I}f_i$, where $\psi$ is a function on V of class $C^{\infty}$.

c) Let T be a torus, $(\alpha_i)_{i\in I}$ a finite family of characters of T distinct from 1, such that the $K_i=$ Ker $\alpha_i$ are pairwise distinct. Then, a function $\varphi$ on T of class $C\prod^{\infty}$ vanishes on the union of the $K_i$ if and only if it can be written as $\psi ._{i\in I}(\alpha_i-1)$, where $\psi$ is a function of class $C^{\infty}$ on T (argue locally on T

and use $b))$.

d) With the notations of no. 4, prove that the map $b_{\infty}:\mathscr{C}^{\infty}(T)^W\rightarrow \mathscr{C}^{\infty}(T)^{-W}$ is bijective.