Uncountable infinite dimensional Hilbert space

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SUMMARY

The discussion centers on the existence of uncountable infinite-dimensional Hilbert spaces, with specific references to the Banach space \(L_{\infty}\) and the space \(\mathcal{H}\) of complex-valued functions defined on \(\mathbb{R}\). Participants highlight that while \(L_{\infty}\) has uncountable dimensions, it is not a Hilbert space. The example provided by L. Debnath and P. Mikusinski illustrates a non-separable Hilbert space relevant to quantum field theory, specifically Fock space, which is noted for its importance in handling variable particle numbers. The conversation also touches on the implications of separability in quantum mechanics.

PREREQUISITES
  • Understanding of Hilbert spaces and their properties
  • Familiarity with functional analysis concepts, particularly Banach spaces
  • Knowledge of quantum mechanics and the role of Fock space
  • Basic grasp of mathematical proofs involving countability and uncountability
NEXT STEPS
  • Study the properties of non-separable Hilbert spaces in quantum field theory
  • Explore the mathematical foundations of Fock space and its applications
  • Learn about the implications of Haag's theorem in quantum mechanics
  • Investigate the differences between separable and non-separable spaces in functional analysis
USEFUL FOR

Mathematicians, physicists, and students of quantum mechanics interested in the theoretical underpinnings of Hilbert spaces and their applications in quantum field theory.

  • #31
When I had to deal with non-seperable Hilbert spaces, decades ago, all interested students
knew the (in a sense trivial) example cited here by dextercioby and the non-trivial example of 'almost-periodic functions'. Also everybody knew that the Fock space over a seperable Hilbert space (acting as 'one-particle space') is separable. Today I 'know' that all Hilbert spaces are finite-dimensional! Tempora mutantur!
 

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