I would like to prove that the Legendre polynomials form a complete basis on the interval [-1, 1] using functional analysis.
Here is what I came up with so far. Legendre polynomials $P_n(x)$ are solutions of the following Sturm–Liouville problem:
$$-\frac{d}{dx}\left[p(x)\frac{du}{ dx}\right]+q(x)u=\lambda w(x)u$$
with $p(x)=1-x^2$, $q(x)=0$, $w(x)=1$ and $\lambda=n(n+1)$. This can be written as:
$$L u = \lambda u$$
where
$$L u = {1 \over w(x)} \left(-{d\over dx}\left[p(x){du\over dx}\right]+q(x)u \right)$$
Using proper boundary conditions, it can be proven that $L$ is self-adjoint. The eigenvalues are $n(n+1)$. They are discrete. One can then easily prove that the eigenvectors ($P_n(x)$) are orthogonal.
Questions:
1) What else do I need to prove to show that the spectrum is discrete?
2) What exactly do I need to prove to show that these eigenvectors form a complete basis? What mathematical theorem/theorems can be used?
3) How would 2) change if some part of the spectrum was continuous?