Orthonormal Subset/Examples/L-2 Space over Interval of Zero to Two Pi

Example of Orthonormal Subset
Let $L^2_\C \closedint 0 {2 \pi}$ be the complex $L^2$ space over the closed interval $\closedint 0 {2 \pi}$.

Let $\innerprod \cdot \cdot$ be the $L^2$ inner product.

For $n \in \Z$, let $e_n: \closedint 0 {2 \pi} \to \C$ be defined by:


 * $\map {e_n} t = \paren{ 2 \pi }^{-1/2} \map \exp {i n t}$

Then $\set{ e_n : n \in \Z}$ is an orthonormal subset of $L^2_\C \closedint 0 {2 \pi}$.