%% ==intuition ## rigour ## What is Mascheke’s theorem? Can you outline the proof? supppose G is a finite group, and V is a finite-dimensional (really just needs to be compact) representation of G → ℝ or ℂ. then every invariant subspace W of V has an invariant complement W′.
suppose ∃ an invariant Hermitian product on V. Then Mascke’s theorem follows from lemma 4.10 and we take W⟂. We need to find an invariant Hermitian product on V. We do this by averaging. Start with any Hermitian product on V, ⟨,⟩ and take $\left< u, v \right>_{new} = \frac{1}{\left| G \right| } \sum _{g\in G}\left< g\cdot v , g\cdot w \right>$. We claim that this is a Hermitian product that is invariant under G
%% ==exam clinic %% ==examples and non-examples %% ==related tags math