Let ($χ,V$) be a irreducible representation and $h$ be class function. Then, why $\sum_{g∈G}\overline {h(g)}χ_v(g)$ is scalar of identity map of $V$ ?
(I try to prove find the relation between $\sum_{g∈G}\overline {h(g)}χ_v(g)$ and $J＝$$\sum_{g∈G}\overline {χ(g)}γ_v(g)$（$γ_v$ is character of ($χ,V$)). The latter is scalar, and I want to prove the former is multiple of $Id_V$ by scalar using the latter.)
And I want to know why the scalar is J｜G｜/dimV.
Thank you in advance.
P.S. Exsitence of scalar is guaranteed by Shur's lemma. Thank you for a comment.