The following is the proposition 3.3 of folland "A Course in Abstract Harmonic Analysis" book. please Explain its proof in more details:
I do not know the cause of contradiction. that is, how maximality is violated?
2026-04-05 18:35:47.1775414147
Explain this proof in more details
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Suppose $u$ is orthogonal to all $M_a$s. Then the cyclic rep $U$ generated by $u$ is orthogonal to the $M_a$s.
Therefore the collection $\{M_a\}_{a\in\cal A}\cup\{U\}$ is also a collection of mutually orthogonal cyclic reps, and it is strictly greater than the collection $\{M_a\}_{a\in\cal A}$. (The partial order here is inclusion.) But our hypothesis was that $\{M_a\}_{a\in\cal A}$ was the greatest possible collection of mutually orthogonal cyclic reps. How can it be the greatest possible when there is one greater? It can't be - that's a contradiction!