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I am looking for a short introduction to nuclear spaces and nuclear operators. I am interested in these spaces as they often arise in mathematically rigorous quantum field theories. I have read the Wikipedia but I'm still having trouble understanding what they are and why we care about them. Most of the other resources I've come across are written for experts in functional analysis, which I am not.

These lecture notes by Paul Garrett are the only resource I've been able to find so far that similar to what I'm looking for.

CBBAM
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  • Nuclear spaces are discussed in many functional analysis books (e.g. Grothendieck, Schaefer, Trèves), as well as "along the way" in a number of books on mathematical QFT. This is why I think you would need to be a bit more specific -- is there anything concrete that you don't understand? –  Jan 12 '24 at 12:49
  • @B.P. I have been trying to understand the projective tensor product approach to defining nuclear spaces and nuclear operators between Banach spaces, but it seems like this isn't needed as most QFT applications only speak of nuclear spaces in the context of Hilbert spaces. Gelfand's book takes this approach and is what I am currently working through. Is this enough for most of the QFT applications? If not, are there any QFT books you can recommend that build up nuclear spaces? – CBBAM Jan 12 '24 at 17:49
  • Maybe I should also ask what "flavour" of QFT you are interested in? If it is constructive QFT, then nuclear spaces will mostly be relevant because spaces of distributions are nuclear spaces (cf. Osterwalder-Schrader), and the most useful definition is in terms of cofiltered limits of Hilbert spaces along trace class maps (leading e.g. to Bochner-Minlos). I think that's the approach that Paul Garrett takes in the notes you mentioned, for example (if I remember correctly). If you are interested in functorial QFT and topological tensor products, the story is quite different, I guess. –  Jan 16 '24 at 18:40
  • @B.P. I am most interested in constructive QFT. – CBBAM Jan 17 '24 at 03:48
  • The best short intro to nuclear locally convex (vector) spaces (lcs) is the little book by A. Pietsch, "Nuclear Locally Convex Spaces". Pietsch was the pioneer of the approach to nuclear lcs using operator ideals instead of modelling the former on the theory of topological tensor products as Grothendieck originally did in his PhD thesis. That being said, this is a very terse book which still requires previous knowledge of functional analysis and maybe Pietsch's approach to nuclear lcs isn't the one you seek... Btw, Glimm-Jaffe's book on constructive QFT also briefly discusses nuclear lcs. – Pedro Lauridsen Ribeiro Mar 10 '24 at 05:03
  • @PedroLauridsenRibeiro Thank you for the references. As a related question, do you have any resource recommendations for learning constructive QFT? I am following Glimm-Jaffe's book which is great but it would be nice to have an alternative (and easier to follow) text to supplement it. – CBBAM Mar 10 '24 at 17:17
  • See this MO answer of mine: https://mathoverflow.net/a/155122/11211

    That being said, none of these references really are any easier than Glimm-Jaffe, they complement and help the latter but don't actually circumvent it. Constructive QFT is a difficult and highly technical subject with lots of open problems, and the the available textbooks on the subject just reflect that state of affairs.

     – Pedro Lauridsen Ribeiro Mar 10 '24 at 19:46
  • @PedroLauridsenRibeiro Thank you very much! – CBBAM Mar 10 '24 at 22:31

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