I'm working on proving whether a representation is unitarisable, so I need to show that for each matrix that my representations maps a group element to that $\langle \varphi (g) v | \varphi (g) w \rangle = \langle v | w \rangle $.
I know that given a matrix U which is unitary it holds that $\langle U v | U w \rangle = \langle v | w \rangle $. Where this is the standard Hermitian inner product. What I'm trying to figure out right now is how to get a matrix operator such that for a given inner product $\langle v | w \rangle _H $, we know that UU' = I implies that $\langle U v | U w \rangle _H = \langle v | w \rangle _H $
A comment on this post Unitary Matrices and the Hermitian Adjoint seems to say that the conjugate transpose has this implication for all inner products. I've tried to find a proof online of the equivalence but couldn't. So any help with that proof or insight into ways to prove that a representation isn't uniterisable would be very appreciated.