I want to prove that if $D$ is a hermitian operator, then $D^\*$ is also a hermitian operator.

$D$ is a hermitian operator implies that $(f,Dg) = (Df,g)$ where $f$ and $g$ are functions.

Therefore, I want to prove that $(f,D^\*g) = (D^\*f,g)$. 

I know that $(f,D^\*g) = (D^\*g,f)^\* = (\int\_a^bDg^\*f\mathrm{d}x)^\*$. I suppose that $(\int\_a^bDg^\*f\mathrm{d}x)^\* = \int\_a^bD^\*gf^\*\mathrm{d}x$. 

Nevertheless, I don't know how I will be able to have $D^\*$ act on $f$ instead of acting on $g$, which appear to be necessary in order to complete the proof...