Question: Prove that a normal ........  Edit

Answer: If T is self adjointi,e.T*=T, then for all v, (⟨Tv,v⟩) ̅ = ⟨v,Tv⟩ = ⟨T*v,v⟩ = ⟨Tv,v⟩., Hence ⟨Tv,v⟩ is equal to its complex conjugate , so that it is real.
Conversely, If ⟨Tv,v⟩ is real for all v, then ⟨Tv,v⟩ =(⟨Tv,v⟩) ̅ = (⟨v,T*v⟩) ̅ = ⟨T*v,v⟩ . Hence 0= ⟨Tv,v⟩-⟨T*v,v⟩ = ⟨(T-T*)v,v⟩.since this is complex inner product space, it implies T-T* = 0. So, T is self adjoint.
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