The question is given below:
And the other questions mentioned are (I know the solutions of all of them):
Sorry for the bad formulation of the my question at the first time I have edited it
I think I should use this theorem in the proof of the first part:
As I know that $SU_{2}$ is a compact topological group and I know that $\Phi_{n}$ is a series of irreducible complex representation of $SU_{2}$ then their matrix elements form a complete orthogonal set in the space $C_{2}(SU_{2})$ by the theorem where $C_{2}(X)$ denote infinite dimensional hermitian space. My problem is that the question requires the complete orthonormal set in the space of continuous central functions on SU_{2} , could anyone help me in showing this please?
Also for the second part of the question I do not know how to show it from the following givens (especially the three problems the author require me to used), could anyone help me please in this part?