\(\displaystyle L_2[0,pi]\) 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