every Hilbert space has an orthonormal basis
Proof : As could be expected, the proof makes use of Zorn’s Lemma. Let be the set of all orthonormal sets of . It is clear that is non-empty since the set is in , where is an element of such that .
The elements of can be ordered by inclusion, and each chain in has an upper bound, given by the union of all elements of . Thus, Zorn’s Lemma assures the existence of a maximal element in . We claim that is an orthonormal basis of .
Thus, we conclude that , i.e. there are elements which are orthogonal (http://planetmath.org/OrthogonalVectors) to . This contradicts the maximality of since, by picking an element with , would belong belong to and would be greater than .
Hence, , and this finishes the proof.
|Title||every Hilbert space has an orthonormal basis|
|Date of creation||2013-03-22 17:56:05|
|Last modified on||2013-03-22 17:56:05|
|Last modified by||asteroid (17536)|