# Inner product space proof question

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• Apr 18th 2010, 06:53 PM
firebio
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.
• Apr 18th 2010, 07:39 PM
tonio
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.

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

general, $=\overline{}$ , so:

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

Tonio