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Math Help - Inner product space proof question

  1. #1
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    Inner product space proof question

    Let T be a linear operator on a complex inner product space V.
    How to prove that T is Hermitian if and only if <T(x),x> is real for all x in V.

    Dont really have any idea how to start.

    Any help would be appreciated.

    Thanks in advance.
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  2. #2
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    Quote Originally Posted by firebio View Post
    Let T be a linear operator on a complex inner product space V.
    How to prove that T is Hermitian if and only if <T(x),x> is real for all x in V.

    Dont really have any idea how to start.

    Any help would be appreciated.

    Thanks in advance.

    T\,\,\,Hermitian\,\iff T=T^{*}\iff\,\,\forall,u\in V\,,\,\,<Tu,u>=<u,T^{*}u>=<u,Tu> \iff <Tu,u>=<u,Tu>\,\,\,\forall u\in V ; but in

    general, <x,y>=\overline{<y,x>} , so:

    T is Hermitian \iff \forall u\in V\,,\,\,<Tu,u>=<u,Tu>=\overline{<Tu,u>} \iff ... end the argument.

    Tonio
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