Fourier-Legendre Series and Closure

Question

My professor taught us this last week

Legendre Polynomials form a maximal orthogonal set in this larger space (previously we were looking at polynomials)
This means that there is no nonzero square-integrable function which is orthogonal to all Legendre polynomials. This allows us to expand any square-integrable function f(x) on [−1, 1] in a series of Legendre polynomials $$\sum_{n\geq0} c_nP_n(x)$$ where $$c_n = \frac{<f,P_n>}{<P_n,P_n>}$$ This is called the Fourier-Legendre series

My doubt in this is how can we write this? Legendre polynomials don't form a basis in the space of square integrable functions.
Further, the more pressing issue is that we are doing an infinite summation, and vector spaces are closed only under finite additions. What is the guarantee that this sum would still belong to the vector space of square integrable functions?

Answer 1

$L^2[-1,1]$ is a Hilbert space, and the normalized Legendre polynomials $\{ p_n \}_{n=0}^{\infty}$ form a complete orthonormal subset of $L^2$. (Normalized means $\langle p_n,p_n\rangle=1$ for all $n$.) Completeness means that, for any $f\in L^2$, if $\langle f,p_n\rangle=0$ for all $n=0,1,2,3,\cdots$, then $f=0$ a.e.. Completeness is equivalent to $\lim_{N\rightarrow\infty}\sum_{n=0}^{N}\langle f,p_n\rangle p_n=f$ for all $f$, where the limit is taken with respect to the $L^2$ norm.