Note: A "representation" for me is a continuous linear action of a topological group on a complex topological vector space. A "subrepresentation" is a closed action-invariant linear subspace.
Question: Is there an irreducible representation of $U(1)$ which is not one dimensional? An example on a Banach space would be particularly nice.
Effort: Such representation cannot be a representation on a Hilbert space. Maybe we can take $1\leq p\leq\infty,p\neq 2$, consider the action on $L^p(S^1)$ and find some irreducible subrepresentation $V$ (the topology of our chosen $V$ must be a topology which cannot be induced from an inner product, otherwise this will not work).