Definition: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 indexed family of real functions all of which are integrable over the interval $\left({a \,.\,.\, b}\right)$.

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

where $\delta_{m n}$ denotes the Kronecker delta.

Then $S$ is defined as orthonormal.