Representation equivalent to a unitary one

[Yoran91]

Hey guys,

How come a representation $\rho$ of a group $G$ is always equivalent to a unitary representation of $G$ on some (inner product) space $V$ ?

Can anyone provide a good source (book, preferably) which states a proof?

Thanks

 

[Office_Shredder]

I think it's only true for finite groups in general. You keep your representation and just define an inner product:
$$\left< u, v \right> = \frac{1}{|G|} \sum_{g\in G} \left< \rho(g) u, \rho(g) v \right>$$

And it's clear that this inner product is invariant under multiplying u and v by any $\rho(h)$ because the right hand side will still end up being the sum over all group elements of $\rho(gh)$ which still gives every $\rho(g)$ once.

If you had a compact Lie group or something you could do the same thing with integrating over the group

 

[Yoran91]

Ok, so what if I had $SO(3)$ ? It's supposed to hold for this group, but I can't seem to find a source (other than the lecture notes I'm using)