orthonormal basis
Definition
An orthonormal basis^{} (or Hilbert basis) of an inner product space^{} $V$ is a subset $B$ of $V$ satisfying the following two properties:

•
$B$ is an orthonormal set.

•
The linear span of $B$ is dense in $V$.
The first condition means that all elements of $B$ have norm $1$ and every element of $B$ is orthogonal^{} (http://planetmath.org/OrthogonalVectors) to every other element of $B$. The second condition says that every element of $V$ can be approximated arbitrarily closely by (finite) linear combinations^{} of elements of $B$.
Orthonormal bases of Hilbert spaces
Every Hilbert space^{} has an orthonormal basis. The cardinality of this orthonormal basis is called the dimension^{} of the Hilbert space. (This is welldefined, as the cardinality does not depend on the choice of orthonormal basis. This dimension is not in general the same as the usual concept of dimension for vector spaces^{} (http://planetmath.org/Dimension2).)
If $B$ is an orthonormal basis of a Hilbert space $H$, then for every $x\in H$ we have
$$x=\sum _{b\in B}\u27e8x,b\u27e9b.$$ 
Thus $x$ is expressed as a (possibly infinite) “linear combination” of elements of $B$. The expression is welldefined, because only countably many of the terms $\u27e8x,b\u27e9b$ are nonzero (even if $B$ itself is uncountable), and if there are infinitely many nonzero terms the series is unconditionally convergent. For any $x,y\in H$ we also have
$$\u27e8x,y\u27e9=\sum _{b\in B}\u27e8x,b\u27e9\u27e8b,y\u27e9.$$ 
Title  orthonormal basis 
Canonical name  OrthonormalBasis 
Date of creation  20130322 14:02:29 
Last modified on  20130322 14:02:29 
Owner  yark (2760) 
Last modified by  yark (2760) 
Numerical id  19 
Author  yark (2760) 
Entry type  Definition 
Classification  msc 46C05 
Synonym  Hilbert basis 
Related topic  RieszSequence 
Related topic  Orthonormal 
Related topic  ClassificationOfHilbertSpaces 
Defines  dimension of a Hilbert space 
Defines  dimension 