proof of classification of separable Hilbert spaces
The strategy will be to show that any separable, infinite dimensional Hilbert space is equivalent to , where is the space of all square summable sequences. Then it will follow that any two separable, infinite dimensional Hilbert spaces, being equivalent to the same space, are equivalent to each other.
Since is separable, there exists a countable dense subset of . Choose an enumeration of the elements of as . By the Gram-Schmidt orthonormalization procedure, one can exhibit an orthonormal set such that each is a finite linear combination of the ’s.
Next, we will demonstrate that Hilbert space spanned by the ’s is in fact the whole space . Let be any element of . Since is dense in , for every integer , there exists an integer such that
The sequence is a Cauchy sequence because
Hence the limit of this sequence must lie in the Hilbert space spanned by , which is the same as the Hilbert space spanned by . Thus, is an orthonormal basis for .
To any associate the sequence . That this sequence lies in follows from the generalized Parseval equality
which also shows that . On the other hand, let be an element of . Then, by definition, the sequence of partial sums is a Cauchy sequence. Since
|Title||proof of classification of separable Hilbert spaces|
|Date of creation||2013-03-22 14:34:11|
|Last modified on||2013-03-22 14:34:11|
|Last modified by||rspuzio (6075)|