Definition:Complete Orthonormal Set of Real Functions

Definition
Let $I$ be an indexing set.

Let $S := \family {\map {\phi_i} x}_{i \mathop \in I}$ be an orthonormal set of real functions over the interval $\openint a b$.

Let $S$ have the property that:
 * $\forall n \in I: \displaystyle \int_a^b \map \psi x \map {\phi_n} x \rd x = 0 \implies \map \psi x \equiv 0$

for any real function $\psi$ integrable over the interval $\openint a b$.

Then $S$ is a complete orthonormal set of real functions.