Hello

$\displaystyle L_2[0,pi]$ is the Hilbert space of complex square integrable functions on [0,pi] and D={f belongs to $\displaystyle L_{2}[0,pi]$ : f is twice continuously differentiable and f(0)=f(pi)=0)}.

let L :-->$\displaystyle L_2[0,pi]$ be the sturm-Liouville operator $\displaystyle Lf=-(e^xf')'$

show that $\displaystyle <Lf,f>\geq0$ for all f in D. Further, show that all eigenvalues of L are positive.

Can you please give me some hits on how to do this as I have final exam on this topic tomorrow morning.

many thanks