# Physical (Quantum Mechanical) Significance of completeness of Hilbert Spaces.

I'm not sure if the question is very 'mathematical',but I'm asking any way. I have a basic knowledge of quantum mechanics and I'm studying Hilbert spaces.

I was wondering what is the physical significance of the completeness of a Hilbert Space wrt QM. i.e. why is the completeness required?

How do mathematical properties of Hilbert spaces or Banach spaces play a role in Quantum Mechanics?

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One important point is that you can have a countable orthonormal basis. Otherwise, you are stuck with overcountable Hamel bases. –  Raskolnikov Apr 25 '12 at 11:42
Intersting question. I suspect Hilbert spaces are the setting of choice because of mathematical convenience, but I'd like to know more. –  Olivier Bégassat Apr 25 '12 at 12:08
Every non complete space with a inner product is a subspace of a complete one which is it closure, So if you are in a space where all the Cawchy sequence don't converges you may assume that you are in the closure. About the Hilbert basis the pre-Hilbert space has to be separable! Of course I am talking in mathematician view point! –  checkmath Apr 25 '12 at 12:13