TITLE: Proving a family of orthogonal functions is complete over a certain interval
QUESTION [7 upvotes]: I'm reading Applied Partial Differential Equations by DuChateu and Zachmann, and the first couple of chapters contain quite a bit of review of Fourier series, as well as theory about L2 integrable functions and orthogonal/orthonormal basis sets of functions.
Some of the exercises require showing that a particular family of functions is not a complete orthogonal or orthonormal family over a certain interval.  The definition given by the book which I am able to remember is that a family of functions is not complete in L2 if $\exists$ a non-zero function $g \in L^2: (g, u_k) = 0$, $\{u_k: k = 1, 2,\ldots\}$  The full definition of a complete family of functions is here.
There don't appear to be any exercises on how to prove a family of functions is complete; it would seem this is a more difficult task.  For example, the family of functions $(\frac{2}{\pi})^{\frac{1}{2}}\sin(kx)$ is stated to be complete and orthonormal on the interval $L^2(0, \pi)$; IIRC the family of function $\sin(\frac{n\pi x}{L})$ is complete and orthogonal (but not orthonormal) on the interval $(0, L)$.
I don't think the definition in 2 would be much help, certainly one can't evaluate every possible piecewise continuous function on the interval in question? The (perhaps simpleminded) approach I had in mind (assuming the converse of the the above definition of a function not being complete is true) would be to go from there with a proof by contradiction, or else somehow show that if there is some function g that purports to make the inner product $(g, u_k)$ zero on the interval, that this function by necessity would have to converge to $u_j, j \neq k$ at all points on the interval to make the inner product integral go to zero.  I haven't gotten very far in my attempts, though, and any further advice on how to go about this would be appreciated.

REPLY [7 votes]: The usual way to prove that families of functions are complete in this sense is via the Stone-Weierstrass theorem, a proof of which can be found here. Essentially this states that a family of real-valued functions $A$ over a compact Hausdorff space $S$ is complete if the following conditions hold:

$A$ is an algebra, i.e.

$\forall f, g \in A: fg \in A$
$\forall f, g \in A: f + g \in A$
$\forall \lambda \in \mathbb{R}, f \in A: \lambda f \in A$
(all these operations are defined pointwise, i.e. $fg(x) = f(x)g(x)$, etc)

$A$ contains all constant functions, i.e.

$\forall \lambda \in \mathbb{R} : \exists f \in A \quad f(x) = \lambda \forall x$ 

$A$ separates points, i.e.

$\forall x, y \in S :  \exists f \in A \quad f(x) \ne f(y)$


These conditions hold for a large number of commonly used families of functions, including polynomials and trigonometric functions.