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

Proof

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Sources

• 1990: John B. Conway: A Course in Functional Analysis (2nd ed.) ... (previous) ... (next): $\text{I}$ Hilbert Spaces: $\S 4.$ Orthonormal Sets of Vectors and Bases: Example $4.3$