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

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

Proof

This theorem requires a proof. In particular: "It is also true that $\set{e_n: n \in \N}$ is a basis, but this is best proved after a bit of theory" You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by crafting such a proof. To discuss this page in more detail, feel free to use the talk page. When this work has been completed, you may remove this instance of {{ProofWanted}} from the code. If you would welcome a second opinion as to whether your work is correct, add a call to {{Proofread}} the page.

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