# Show that the characters of the representations $\phi_{n}$ of $SU(2)$ constitute a complete orthogonal set.

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?

• What is 7.4? which book is this from? – Sheve Apr 6 at 15:13
• Ernest B. Vinberg ..... "Linear representations of groups "@Sheve – hopefully Apr 6 at 15:20
• – hopefully Apr 6 at 15:41
• @Sheve and this is a solution ofanother one problem of the problems mentioned math.stackexchange.com/questions/3168577/… – hopefully Apr 6 at 22:41
• * this follows immediately since all trace functions (characters) are central – Sheve Apr 13 at 8:57