We have a Sturm-Liouville operator $$ L=\frac{1}{w(x)}\left[\frac{d}{dx}\left(p(x)\frac{d}{dx}\right)+q(x)\right] $$ and consider $$ \frac{\partial c}{\partial t}=Lc, $$ with homogeneous boundary conditions.

If we are now searching for solutions, the technique is to start with considering the homogeneous equation, i.e. $q(x)=0$ and solves the Eigenvalue problem $$ L\Phi=\lambda\Phi. $$ There are functions $\Phi$ - called eigenfunctions - that solve this eigenvalue problem, they exist since $L$ is self-adjoint under homogeneous boundary conditions. The eigenfunctions are orthogonal.

Moreover, the eigenfunctions $\Phi$ form a basis for the function space consisting of functions that satisfy the boundary conditions, meaning that any such function can be expressed as a linear combination of the eigenfunctions. So we can find solutions of the inhomogeneous equation by making the approach $u(x,t)=\sum_n A_n\Phi_n$, put this into the equation and determine the constants.

**My question is** how one can show/ see that the eigenfunctions form a basis of the function space consisting of functions that satify the boundary conditions.

More precisely, I think, the function space for which the eigenfunctions form a basis is supposed to be the function space containing all functions that

(i) are quadrat-integrable with respect to the weight function $w$ and

(ii) satisfy the boundary conditions.

*Do not know exactly if we really need (i).*

Wikipedia says that the proper setting is the Hilbert space $L^2([a,b], w(x)dx)$ and that in this space, $L$ is defined on sufficiently smooth functions that satisfy the boundary conditions.

Anyhow: How to show/ see that the eigenfunctions form a basis?