Proof that every Eigenvalue of self adjoint operator is real

 Theorem: Every Eigenvalue of self adjoint operator is real

\[\begin{array}{l}\\ Suppose\;T\;is\;self\;adjoint \Rightarrow T = {T^ * }\\\\ \;Let\;\lambda \in F\;\;\;be\;eigenvalue\;of\;T\\\\ \Rightarrow \exists \;v \in \;{\cal V}\;s.t\;\;\;Tv = \lambda v\\\\ note\;that:\lambda {\left\| v \right\|^2} = \lambda \left\langle v \right.,\left. v \right\rangle \\\\ = \;\langle \lambda v,v\rangle \\\\ = \;\langle Tv,v\rangle \\\\ = \;\langle v,{T^ * }v\rangle \\\\ = \;\langle v,Tv\rangle \\\\ = \;\langle v,\lambda v\rangle \\\\ = \;\bar \lambda \langle v,v\rangle = \;\bar \lambda {v^2}\\\\ \lambda = \;\bar \lambda \to \lambda \;is\;real\\ \end{array}\]