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every Hilbert space has an orthonormal basis
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(Theorem)
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Theorem - 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  .
It is clear that  is an orthonormal set, as it belongs to
 . It remains to see that the linear span of  is dense in  .
Let
 denote the closure of the span of  . Suppose
 . By the orthogonal decomposition theorem we know that
Thus, we conclude that
 , i.e. there are elements which are orthogonal 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. 
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(view preamble)
Cross-references: orthogonal decomposition theorem, closure, dense in, linear span, maximal element, union, upper bound, chain, inclusion, clear, orthonormal sets, Zorn's lemma, orthonormal basis, Hilbert space
There is 1 reference to this entry.
This is version 1 of every Hilbert space has an orthonormal basis, born on 2008-03-21.
Object id is 10431, canonical name is EveryHilbertSpaceHasAnOrthonormalBasis.
Accessed 247 times total.
Classification:
| AMS MSC: | 46C05 (Functional analysis :: Inner product spaces and their generalizations, Hilbert spaces :: Hilbert and pre-Hilbert spaces: geometry and topology ) |
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Pending Errata and Addenda
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