Inner product in Maschke's Theorem I am working through Maschke's Theorem on page 16 in Bruce Sagan's The Symmetric Group:
In order to prove the theorem the author constructs an inner product $\langle v, w \rangle' = \sum_{g \in G} \langle gv , gw \rangle$.
However, I fail to see why $\langle v, w \rangle'$ satisfies the definiteness property of the inner product. In other words, I do not know how to prove $\langle v,v \rangle = 0$ if and only if $v = 0$.
The "if"-implication is obvious, but I cannot see why $v = 0$ if $\langle v, w \rangle' = 0$.
Thank you very much for your help!

If $ \langle v, v \rangle' = \sum_{g \in G} \langle gv , gv \rangle = 0$ but each of the terms are non-negative so they must all be zero... $ \langle gv , gv \rangle = 0$ implies that $v = 0$.
Hmm... I wonder if you could say the average of a bunch of positive definite matrices is positive definite.
