classification of Hilbert spaces
Hilbert spaces can be classified, up to isometric isomorphism, according to their dimension. Recall that an isometric isomorphism of Hilbert spaces is an unitary transformation, therefore it preserves the vector space structure along with the inner product structure (hence, preserving also the topological structure). Recall also that the dimension of a Hilbert space is a well defined concept, i.e. all orthonormal bases of an Hilbert space share the same cardinality.
The classification theorem we describe here states that two Hilbert spaces and are isometrically isomorphic if and only if they have the same dimension, i.e. if and only if an orthonormal basis of has the same cardinality of an orthonormal basis of .
This will be achieved by proving that every Hilbert space is isometrically isomorphic to an space (http://planetmath.org/EllpXSpace), where has the cardinality of any orthonormal basis of the Hilbert space in consideration.
Theorem 1 - Suppose is an Hilbert space and let be a set that indexes one (and hence, any) orthonormal basis of . Then, is isometrically isomorphic to .
Theorem [Classification of Hilbert spaces] - Two Hilbert spaces and are isometrically isomorphic if and only if they have the same dimension.
Proof of Theorem 1: Let an orthonormal basis indexed by the set . Let be defined by
We claim that is an isometric isomorphism. It is clear that is linear. Using Parseval’s equality and the definition of norm in it follows that
We conclude that is isometric. It remains to see that it is surjective (since injectivity follows from the isometric condition).
or in other , . Hence, is surjective.
Proof of the classification theorem :
Of course, if the Hilbert spaces and are isometrically isomorphic, with isometric isomorphism , then if is an orthonormal basis for than is an orthonormal basis for . Hence, and have the same dimension.
If the Hilbert spaces and have the same dimension, then we can index any orthonormal basis of and any orthonormal basis of by the same set . Using Theorem 1 we see that and are both isometrically isomorphic to . Hence and are isometrically isomorphic.
|Title||classification of Hilbert spaces|
|Date of creation||2013-03-22 17:56:18|
|Last modified on||2013-03-22 17:56:18|
|Last modified by||asteroid (17536)|
|Synonym||Hilbert spaces of the same dimension are isometrically isomorphic|
|Defines||every Hilbert space is isometrically isomorphic to a space|