Proof that every Eigenvalue of self adjoint operator is real

 Theorem: Every Eigenvalue of self adjoint operator is real

$$\begin{array}{l}\\ SupposeTisselfadjoint\Rightarrow T=T^{\ast }\\ \\ Let\lambda \in FbeeigenvalueofT\\ \\ \Rightarrow \mathrm{\exists }v\in \mathcal{V}s.tTv=\lambda v\\ \\ notethat:\lambda {\Vert v\Vert }^2=\lambda ⟨v,v⟩\\ \\ =⟨\lambda v,v⟩\\ \\ =⟨Tv,v⟩\\ \\ =⟨v,T^{\ast }v⟩\\ \\ =⟨v,Tv⟩\\ \\ =⟨v,\lambda v⟩\\ \\ =\overline{\lambda }⟨v,v⟩=\overline{\lambda }{v}^2\\ \\ \lambda =\overline{\lambda }\to \lambda isreal\end{array}$$