orthonormal basis


An orthonormal basisMathworldPlanetmath (or Hilbert basis) of an inner product spaceMathworldPlanetmath V is a subset B of V satisfying the following two properties:

The first condition means that all elements of B have norm 1 and every element of B is orthogonalMathworldPlanetmath (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 combinationsMathworldPlanetmath of elements of B.

Orthonormal bases of Hilbert spaces

Every Hilbert spaceMathworldPlanetmath has an orthonormal basis. The cardinality of this orthonormal basis is called the dimensionPlanetmathPlanetmath 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 spacesMathworldPlanetmath (http://planetmath.org/Dimension2).)

If B is an orthonormal basis of a Hilbert space H, then for every xH we have


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,bb 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,yH 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