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As far as I understand the postulates of quantum mechanics use only properties of abstract Hilbert space. So could we use any other Hilbert space for calculations instead of $L_2$? What could it be?

I can only suggest infinite-dimensional vector space with Euclidean norm, but it seems to be isomorphic to $L_2$.

Qmechanic
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nvvm
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    Hilbert spaces go by Hilbert dimension. You have a unique (up to isomorphism) Hilbert space for any cardinal. – Phoenix87 May 23 '15 at 01:28
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    Isn't Hilbert space essentially $L_2$ by definition? Are you thinking of Banach spaces? – Peter Shor May 23 '15 at 01:31
  • Yes, it is. I thought about continuous spectrum of coordinate operator that is a basis in $L_2$ and morphism of it to space with discrete basis. – nvvm May 23 '15 at 01:41
  • http://en.wikipedia.org/wiki/Hilbert_space#Hilbert_dimension –  May 23 '15 at 04:20
  • Possible duplicates: http://physics.stackexchange.com/q/41719/2451 and links therein. – Qmechanic May 23 '15 at 09:38

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