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.

Quote:

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Originally Posted by firebio

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

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

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

Tonio