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
if , the sequence of partial sums of
is also a Cauchy sequence, so converges
and its limit lies in . Hence the operator is invertible
and
is an isometry between and .
Title | proof of classification of separable Hilbert spaces |
---|---|
Canonical name | ProofOfClassificationOfSeparableHilbertSpaces |
Date of creation | 2013-03-22 14:34:11 |
Last modified on | 2013-03-22 14:34:11 |
Owner | rspuzio (6075) |
Last modified by | rspuzio (6075) |
Numerical id | 5 |
Author | rspuzio (6075) |
Entry type | Proof |
Classification | msc 46C15 |
Related topic | VonNeumannAlgebra |