Inner product space proof question

Quote:

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.

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