Expansion and orthogonality

Question

It is stated various places that as the Legendre functions $$ \sum_{m=0}^\infty P_m(\cos\theta) $$ are orthogonal, this enables us to expand any function of $\theta$ into a set of these functions.

How come?

Answer 1

It's not orthogonality that lets us expand. What orthogonality does is lets us expand easily, because the coefficients for a function $f$ must be exactly

$$ c_m = \langle f, P_m(\cos \theta) \rangle $$

The factor the a function $f$ can be expanded at all comes because these functions form a basis for the set of all functions. (Except that's not quite true: you need to limit to integrable functions, and they probably need to be continuous almost everywhere, and equality holds only on a set whose complement has measure zero, and ... a whole lot of other technical constraints.) More particularly, the existence of an expansion comes from the functions spanning the relevant space of function. The uniqueness of the expansion comes because they're a basis. Orthogonality just makes writing down the coefficients easy.