Basis (Hilbert Space)/Examples

Examples of Basis (Hilbert Space)

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

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

Real Vector Space

Let $\R^d$ be the real vector space with $d$ dimensions.

Let $e_1, \ldots, e_d$ be the standard basis.

Then $\set{ e_1, \ldots, e_d }$ is a basis for the Hilbert space $\R^d$.

Space of Square Summable Mappings

Let $I$ be a set.

Let $\map {\ell^2} I$ be the space of square summable mappings over $I$.

For $i \in I$, define $e_i: I \to \GF$ as:

$\map {e_i} j := \begin{cases}

1 &: i = j \\ 0 &: i \ne j \end{cases}$

Then $\set{ e_i : i \in I}$ is a basis for $\map {\ell^2} I$.