orthonormal basis
Definition
An orthonormal basis (or Hilbert basis) of an inner product space is a subset of satisfying the following two properties:
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•
is an orthonormal set.
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The linear span of is dense in .
The first condition means that all elements of have norm and every element of is orthogonal (http://planetmath.org/OrthogonalVectors) to every other element of . The second condition says that every element of can be approximated arbitrarily closely by (finite) linear combinations of elements of .
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 is an orthonormal basis of a Hilbert space , then for every we have
Thus is expressed as a (possibly infinite) “linear combination” of elements of . The expression is well-defined, because only countably many of the terms are non-zero (even if itself is uncountable), and if there are infinitely many non-zero terms the series is unconditionally convergent. For any we also have
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 |