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In applications, Hilbert spaces of interest are often assumed to be separable. In addition to being extremely convenient mathematically, this assumption can often be justified on computational or physical grounds.

Are there applications where non-separable Hilbert spaces naturally arise?

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The main example of a non-separable Hilbert space is the Besicovitch space of almost periodic functions. Almost periodic functions play a significant role in analysis, from differential equations to operator algebras, and this space is quite useful.

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  • $\begingroup$ Here's Besicovich book (I was giving the same answer...) plouffe.fr/simon/math/… $\endgroup$ – Pietro Majer Aug 8 '13 at 7:19
  • $\begingroup$ @PietroMajer, it says Not Found when you click on the link. $\endgroup$ – Todd Trimble Dec 31 '18 at 18:21
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Here is an application from Machine Learning.

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    $\begingroup$ I can only access the abstract but the abstract do not hint at the use of nonseparable Hilbert spaces. Can you elaborate? $\endgroup$ – Dirk Dec 31 '18 at 13:12
  • $\begingroup$ @Dirk Putting aside the article, it seems that Neuromath is suggesting the theory of RKHS as it applies to machine learning theory is an example. Although it tends to be glossed over, in the theory of kernel methods Moore–Aronszajn's theorem (which is stated for non-separable Hilbert spaces) is useful. Practically, the role of non-separable Hilbert spaces is debatable here still. $\endgroup$ – Josiah Park Dec 31 '18 at 14:39
  • $\begingroup$ On a tangential note, Aronszajn is mentioned heavily in one part of the biography of Grothendieck that is being put together here: webusers.imj-prg.fr/~leila.schneps/grothendieckcircle/…. Grothendieck visited him for some time in Kansas around the time his interests were shifting from functional analysis to algebraic geometry. $\endgroup$ – Josiah Park Dec 31 '18 at 14:51
  • $\begingroup$ A link to the PDF is here. @Josiah Park: yes, I was referring to the use of RKHSs in machine learning theory. The ones used in practice are all separable; one thing this paper shows is that non-separable RKHs don't have the nice properties for machine learning that separable ones do (so not sure if you would really call it an application then! But at least a consideration). $\endgroup$ – Neuromath Dec 31 '18 at 16:00

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