About 20 years ago I read in textbook that "all irreducible representations of compact groups are finite-dimensional", but me and the proof of this fact never met each other :)
May I ask dear MO colleagues, is there (simple?) argument to prove it ?
As far as I heard this result can be generalized in the realm of non-commutative geometry, Woronowicz compact quantum groups (?).
So the "bonus" question - what is appropriate "compactness" condition for some algebra (and/or Hopf algebra) such that it will guarantee the same property (i.e. all irreps are finite-dim.) ?
[EDIT] Thanks very much for excellent answers ! Let me ask about some more details, to finally clarify.
1) What is maximal possible relaxation of the requirement on vector space V ? Is it enough to require arbitrary linear topological space or we need to restrict to Hausdorff (?) Banach (?) Hilbert (?), whatever spaces ? (It seems restrictions on the space may come from the Schur lemma, it is not clear for me what is appropriate generality it holds).
2) Do we need axiom of choice here ? (Probably not, we need existence of Haar measure, but Wikipedia writes that "Henri Cartan furnished a proof of existence of Haar measure which avoided AC use."
3) Informally: what is the hardest tool one uses in the proof ? (May be existence of Haar measure ?)
Let me add sketch of arguments by Aakumadula, as I understand it. It might be helpful to clarify new questions.
1) Tool: Continuous functions on the group can be mapped to operators on V. (Need measure here). (Group algebra acts on V).
2) Fact: Continuous function will be mapped to COMPACT operators. (In R^n I know how to prove it, in general no).
3) Observe: Conjugation invariant function are mapped to operators which commute with action of group.
4) Schur Lemma: operators commuting with group in irrep are Lamda*Id. (What do we need from the space V for this to be true ? )
5) Corollary: If we find invariant continuous function which is mapped in NON-zero in V, then we are done, because by (2) it is compact operator and by (4) it is Lambda*Id.
So we need to find invariant function which will be non-zero in V.
6) Take arbitrary "approximate identity" i.e. sequence of continuous (non-invariant) functions f_n which converge as functionals to delta-function in identity of the group. (It is local fact. But how to prove it ? Do we need Axiom of choice here ? )
7) Make averaging over the group of f_n - get sequence of INVARIANT continuous functions which again converge to detla(e), since delta(e) is invariant.
8) Operators T(f_n) converge to identity operator, hence for some N they are NON-ZERO. WE ARE DONE by (5) ! Because T(f_N) is compact and Lambda*Id and Lambda is NON-ZERO.