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)