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nLab > Latest Changes: elementary treatment of Hilbert spaces

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    • CommentRowNumber1.
    • CommentAuthorAndrew Stacey
    • CommentTimeDec 15th 2009

    I've started a page an elementary treatment of Hilbert spaces. The intention is to see how much of (simple) Hilbert space theory can be done without using the phrases "As a Hilbert space is a normed vector space ..." or "As a Hilbert space is a metric space ...".

    I haven't gotten very far yet, as can be seen! Also, it's not intended to be Deep Mathematics (there's a mild centipedal justification on the page) but just playing with some ideas and trying to see what a Hilbert space really is.

    • CommentRowNumber2.
    • CommentAuthorUrs
    • CommentTimeDec 15th 2009

    I slightly Lab-elved the page, editing the toc and insering some links

    • CommentRowNumber3.
    • CommentAuthorAndrew Stacey
    • CommentTimeDec 15th 2009

    Thanks!

    I should have said: I hope that this is okay for the n-lab.

    • CommentRowNumber4.
    • CommentAuthorUrs
    • CommentTimeDec 15th 2009
    This comment is invalid XHTML+MathML+SVG; displaying source. <div> <blockquote> I hope that this is okay for the n-lab. </blockquote> <p>Most certainly. Why are you worried?</p> </div>
    • CommentRowNumber5.
    • CommentAuthorAndrew Stacey
    • CommentTimeDec 15th 2009

    It didn't have an 'n' in it!

    I wasn't particularly worried, otherwise I would have asked before I started writing, but I suffer from the British disease of always worrying that I'm in the wrong place at the wrong time.

    • CommentRowNumber6.
    • CommentAuthorUrs
    • CommentTimeDec 15th 2009

    Well, I think thi stuff is very good. It doesn't need to have an n in it to be part of the nLab.

    • CommentRowNumber7.
    • CommentAuthorTobyBartels
    • CommentTimeDec 15th 2009

    Yeah, you just have to accept that we'll start remarking on what strength of logic your elementary treatment requires and which categories it can be internalised in, stuff like that.

    • CommentRowNumber8.
    • CommentAuthorAndrew Stacey
    • CommentTimeDec 18th 2009

    @Toby, go ahead!

    Seriously, so long as it's clear which bits are meant as the "elementary" bits and which are commentary, then I would welcome such. As you can see, I'm putting in remarks linking the elementary treatment to the standard treatment. In case it's not clear, the idea of such an elementary treatment is that it be a gateway to the more developed theory but one that is, perhaps, simpler to reach. Without the commentary there wouldn't be any indication of where someone could go once they'd gotten through the gateway. So I see remarks like that as sort of "If you liked this bit, you'll also like ...".

  9. A Hilbert space can be seen as a member of a hierarchy of structures that start very simple and extend with simple additions into more complicated levels of the target structure. The extensions are quite comprehensible, but mathematics imposes severe restrictions to the extension, such that the higher level structure evolves in a particular direction. The hierarchy currently exists of a Hilbert lattice. a Hilbert space and a Hilbert repository. The first two already occur in nlab. A Hilbert repository is a system of quaternionic Hilbert spaces that all share the same underlying vector space. A huge number of separable Hilbert spaces are member of the Hilbert repository. One of them is infinite dimensional and acts as a background platform. It owns a unique non-separable companion Hilbert space that embeds its separable partner. This system of Hilbert spaces forms a huge archive that can easily cope with everything that exists in physical reality. It can act as a very powerful theory development platform for models of physical reality. The Hilbert repository is treated in https://www.researchgate.net/publication/339744488_Representing_basic_physical_fields_by_quaternionic_fields The restrictions seem to explain the shortlist of elementary particles that are exposed in the Standard Model.

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