Definition:Complete Orthonormal Set of Real Functions

Definition
Let $I$ be an indexing set.

Let $S := \left \langle {\phi_i \left({x}\right)} \right \rangle_{i \mathop \in I}$ be an orthonormal set of real functions over the interval $\left({a \,.\,.\, b}\right)$.

Let $S$ have the property that:
 * $\forall n \in I: \displaystyle \int_a^b \phi \left({x}\right) \phi_n \left({x}\right) \, \mathrm d x = 0 \implies \phi \left({x}\right) \equiv 0$

for any real function $\phi \left({x}\right)$ integrable over the interval $\left({a \,.\,.\, b}\right)$.

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