Let $G$ be a finite abelian group that operates linear on the $\mathbb{C}-$vector space $V$ by \begin{equation} V \times F \longrightarrow V, \quad (f, a) \longmapsto f^a. \end{equation}
Then we define $f^\chi := \{ f \in V \mid f^a = \chi(a) f \} $. Then \begin{equation} V = \bigoplus_{\chi} V^\chi, \end{equation} where $\chi$ runs through all characters from $G$ to $\mathbb{C}$. I managed to proove the existence of such a decomposition, by prooving \begin{equation} f = \frac{1}{\vert G \vert} \sum_\chi \sum_{a \in G} \chi(a)^{-1} f^a, \end{equation} but didn't manage to proof the uniqueness of it. We should use somehow \begin{equation} \sum_\chi \chi(a) = \begin{cases} 0, &for \quad a\neq 0 \\ \vert G \vert &for \quad a = e_G, \end{cases} \end{equation}
but I can't find the answer, yet. Does anyone know how to progress?