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The question is given below:

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And the other questions mentioned are (I know the solutions of all of them):

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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:

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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?

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  • $\begingroup$ What is 7.4? which book is this from? $\endgroup$ – Sheve Apr 6 at 15:13
  • $\begingroup$ Ernest B. Vinberg ..... "Linear representations of groups "@Sheve $\endgroup$ – hopefully Apr 6 at 15:20
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    $\begingroup$ @Sheve math.stackexchange.com/questions/3166964/… $\endgroup$ – hopefully Apr 6 at 15:41
  • $\begingroup$ @Sheve and this is a solution ofanother one problem of the problems mentioned math.stackexchange.com/questions/3168577/… $\endgroup$ – hopefully Apr 6 at 22:41
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    $\begingroup$ * this follows immediately since all trace functions (characters) are central $\endgroup$ – Sheve Apr 13 at 8:57

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