Note 1
Pre-Hilbert Space and Hilbert Space
[Theorem]
For pre-Hilbert space H, there exists Hilbert space H~ that contains pre-Hilbert space H as dense subspace .
When there exist two Hilbert spaces H~1 and H~2 and H is contained in H~1 and H~2 as dense subspace of H, H~1 and H~2 are isomorphic by mapping what H transform H’s elements to itself .
[Explanation]
To be continued
Tokyo May 30, 2008
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