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?
Best Answer
There's an MO post with some good thoughts on the role of completeness from a mathematical point of view. (I especially like Andrew Stacey's idea of "going to a party hosted by the Square Integrables in their posh mansion, but spending the whole time hanging out with the Schwartz family" :-)
With a view to quantum mechanics, note that the Riesz representation theorem (which establishes the bra-ket correspondence) depends on completeness. You can trace the role of completeness in this proof, which uses the orthogonal decomposition theorem, whose proof in turn uses completeness.
The history section of the Wikipedia article on Hilbert spaces also has some relevant information.