all orthonormal bases have the same cardinality
Proof: When is finite-dimensional (as a vector space), every orthonormal basis is a Hamel basis of . Thus, the result follows from the fact that all Hamel bases of a vector space have the same cardinality (see this entry (http://planetmath.org/AllBasesForAVectorSpaceHaveTheSameCardinality)).
We now consider the case where is infinite-dimensional (as a vector space). Let and be two orthonormal basis of , indexed by the sets and , respectively. Since is infinite dimensional the sets and must be infinite.
We know, from Parseval’s equality, that for every
We know that, in the above sum, for only a countable number of . Thus, considering as , the set is countable. Since for each we also have
there must be such that . We conclude that .
Hence, since each is countable, (because is infinite).
An analogous proves that . Hence, by the Schroeder-Bernstein theorem and have the same cardinality.
|Title||all orthonormal bases have the same cardinality|
|Date of creation||2013-03-22 17:56:10|
|Last modified on||2013-03-22 17:56:10|
|Last modified by||asteroid (17536)|
|Synonym||dimension of an Hilbert space is well-defined|