Basis (Hilbert Space)/Examples/L-2 Space over Interval of Zero to Two Pi

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Example of Basis (Hilbert Space)

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

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 a basis for $L^2_\C \closedint 0 {2 \pi}$.