all orthonormal bases have the same cardinality

TheoremMathworldPlanetmath. – All orthonormal bases of an Hilbert spaceMathworldPlanetmath H have the same cardinality. It follows that the concept of dimensionPlanetmathPlanetmathPlanetmath of a Hilbert space is well-defined.

Proof: When H is finite-dimensional (as a vector spaceMathworldPlanetmath), every orthonormal basis is a Hamel basisMathworldPlanetmath of H. Thus, the result follows from the fact that all Hamel bases of a vector space have the same cardinality (see this entry (

We now consider the case where H is infinite-dimensional (as a vector space). Let {ei}iI and {fj}jJ be two orthonormal basis of H, indexed by the sets I and J, respectively. Since H is infinite dimensional the sets I and J must be infiniteMathworldPlanetmath.

We know, from Parseval’s equality, that for every xH


We know that, in the above sum, x,ei0 for only a countableMathworldPlanetmath number of iI. Thus, considering x as fj, the set Ij:={iI:fj,ei0} is countable. Since for each iI we also have


there must be jJ such that fj,ei0. We conclude that I=jJIj.

Hence, since each Ij is countable, IJ×J (because J is infinite).

An analogous proves that JI. Hence, by the Schroeder-Bernstein theorem J and I have the same cardinality.

Title all orthonormal bases have the same cardinality
Canonical name AllOrthonormalBasesHaveTheSameCardinality
Date of creation 2013-03-22 17:56:10
Last modified on 2013-03-22 17:56:10
Owner asteroid (17536)
Last modified by asteroid (17536)
Numerical id 7
Author asteroid (17536)
Entry type Theorem
Classification msc 46C05
Synonym dimension of an Hilbert space is well-defined