Definition: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 indexed family of real functions all of which are integrable over the interval $\openint a b$.

Let $S$ have the property that:
 * $\forall m, n \in I: \ds \int_a^b \map {\phi_m} x \map {\phi_n} x \rd x = \delta_{m n}$

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

Then $S$ is defined as orthonormal.