I'm currently revising representation theory and I'm a bit stuck trying to prove the converse of the above statement.
$(\Rightarrow)$ is straight forward because if $f$ is the character of a representation $\rho$, then it is a direct sum of irreducible representations $q_i$. And $\rho\sim n_1q_1\oplus...\oplus n_kq_k$ then $(f,\chi_{q_i})_G =n_i$ which is a non-negative integer.
For $(\Leftarrow)$ I'm not sure how to do it.
I know that $\chi_{q_i}$ form a basis for the space of class functions. So $f=\sum_{i=0}^{k} c_i\chi_{q_i}$ where $c_i\in \Bbb{C}$
If $(f,\chi_{q_i})_G =n_i \in \Bbb{Z}_{\ge 0}$ then im guessing the $n_i=c_i$ (not sure why using the definition of $(\cdot,\cdot)_G$).
And maybe $\rho$ will be the direct num of $n_i$ copies on each $q_i$ but we haven't actually shown that $f$ is a character yet?
Any help would be appreciated.