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 well-defined, 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

$$x=\sum _{b\in B}⟨x,b⟩b.$$

Thus $x$ is expressed as a (possibly infinite) “linear combination” of elements of $B$. The expression is well-defined, because only countably many of the terms $⟨x,b⟩b$ are non-zero (even if $B$ itself is uncountable), and if there are infinitely many non-zero terms the series is unconditionally convergent. For any $x,y\in H$ we also have

$$⟨x,y⟩=\sum _{b\in B}⟨x,b⟩⟨b,y⟩.$$

Title	orthonormal basis
Canonical name	OrthonormalBasis
Date of creation	2013-03-22 14:02:29
Last modified on	2013-03-22 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