Definition:Basis (Hilbert Space)

This page is about bases in the context of Hilbert spaces. For other uses, see Basis.

It has been suggested that this page be renamed. In particular: Definition:Orthogonal Basis (Hilbert Space) To discuss this page in more detail, feel free to use the talk page.

Definition

Let $H$ be a Hilbert space.

A basis for $H$ is a maximal orthonormal subset of $H$.

Thus, $B$ is a basis for $H$ if and only if for all orthonormal subsets $B'$ of $H$:

$B \subseteq B' \implies B = B'$

Examples

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 $\map {\ell^2} I$.

Linguistic Note

The plural of basis is bases.

This is properly pronounced bay-seez, not bay-siz.

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: Definition $4.1$