The Peter-Weyl theorem asserts that for a compact Lie group $G$ every unitary irreducible representation is necessarily finite-dimensional and any unitary representation is a direct sum of irreducibles.
1) Is not any representation of a compact group (finite or infinite-dimensional) representation of a compact group unitarizable and thus completely reducible? Or does this only hold for finite-dimensional?
2) Now if $G$ is not compact, how exactly does the situation change? If I have an infinite-dimensional unitary representation of a non-compact group, is it also always completely reducible? Also, which noncompact group possess (nontrivial) finite-dimensional unitary representations?