If <h|Qh> = <Qh|h> for all functions h, show that <f|Qg> = <Qf|g> for all f and g.
f,g, and h are functions of x
Q is a hermitian operator
Hints: First let h=f+g, then let h=f+ig
The Attempt at a Solution
(All integral are with respect to x, and go from -∞ to +∞)
<(f+g)|Q(f+g)> = <Q(f+g)|(f+g)>
∫(f*+g*)(Qf+Qg) = ∫(Qf*+Qg*)(f+g)
∫f*Qf + ∫g*Qg + ∫f*Qg + ∫g*Qf = ∫(Qf*)f + ∫(Qg*)g + ∫(Qf*)g + ∫(Qg*)f
<f|Qf> + <g|Qg> + <f|Qg> + <g|Qf> = <Qf|f> + <Qg|g> + <Qf|g> + <Qg|f>
<f|Qg> + <g|Qf> = <Qf|g> + <Qg|f>
And then I'm stuck.