Uncountable infinite dimensional Hilbert space

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An example of an uncountable infinite-dimensional Hilbert space is provided through the space of complex-valued functions that vanish almost everywhere except at a countable number of points, with square-summable values at those points. The discussion highlights the significance of separability in Hilbert spaces, particularly in quantum mechanics, where separable spaces allow for the definition of a complete orthonormal basis. However, nonseparable Hilbert spaces, such as Fock space in quantum field theory, also play a crucial role, despite their complexity. The conversation reveals differing opinions on the implications of separability and closure in defining these spaces, particularly regarding their physical relevance. Overall, the topic underscores the intricate relationship between mathematical properties of Hilbert spaces and their applications in physics.
  • #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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