TITLE: Recognition of an orthonormal complete set in an Hilbert Space
QUESTION [7 upvotes]: I recently came across this exercise:
Let $u_n:[0,1]\to \mathbb R$ be the sequence of functions defined by:
$$(u_n):=\text{sign }(\sin(2^n\pi x)),\qquad n=0,1,2,\dots$$
a)Prove that this set of functions is an orthonormal system in $L^2([0,1])$.
b)Is this set of functions a complete orthonormal system in $L^2([0,1])$?
This was the exercise. Nothing special I agree, but it happened that a few time ago I answered this topic Link. Two question on orthonormal basis in less than a week are too much to stand for my mind, so the question naturally arose:

Is it possible to find general criteria (similar to the one in the link) to recognize an orthonormal basis or at least could you gently give me some references/proofs of similar/related topics?

I'm not looking for answers of the type: $\langle (u_n)_{n\in\mathbb N}\rangle^{\perp}=0,$ there is no orthonormal set in which $(u_n)_n$ is properly contained and stuff like this, but something less known or, how can I say, a condition easy to verify, more practical than the theoretic definitions. Thanks in advance for your replies.
BTW: if anybody wants to try the exercise, he will be welcomed.

REPLY [2 votes]: Regarding point b
Consider the space of piecewise constant functions in the intervals $ [0,\frac{1}{4}]$, $ [\frac{1}{4},\frac{1}{2}]$, $ [\frac{1}{2},\frac{3}{4}]$, $ [\frac{3}{4},1]$.
This space is orthogonal to all $u_n$, for $n>2$. On the other hand these functions cannot be spanned by the vectors $u_0$, $u_1$, $u_2$, on the basis of dimensionality.
Regarding the general question, here is a criterion valid for reproducing kernel Hilbert spaces:
Let $e_n(x)$, be an orthonormal basis of functions in a reproducing kernel Hilbet space with kernel $k(x,y)$, then:
$\sum_n e_n(x) e_n(y) = k(x,y)$.
(under pointwise convergence). Please see a proof in the following exposition by: V.I. Paulsen (theorem 3.4).