Exercise 1, § 3

(a) Show that every element $u \in A^+(X)$ can be written uniquely in the form $u = \sum_{x \in X} u_x x$, where $u_x \in A(X)$.

(b) Show that $\{0\}$ is the only submodule of $A^+(X)$ which is stable under all the mappings $u \mapsto u_x$ ($x \in X$). (If $a$ is such a submodule and $a \neq \{0\}$, consider a non-zero element of $a$ of minimal degree.)