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nLab > Latest Changes: tensor product of Banach spaces

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    • CommentRowNumber1.
    • CommentAuthorDavidRoberts
    • CommentTimeMay 25th 2018

    Added links to related tensor products, including the new page inductive tensor product (under construction)

    diff, v14, current

  2. There was a question about whether “uniform” was required to land a cross norm between the injective tensor norm and the projective tensor norm. This is, in fact, not a requirement. I have corrected the relevant proposition and provided a reference.

    Evangelos Nikitopoulos

    diff, v15, current

    • CommentRowNumber3.
    • CommentAuthorTobyBartels
    • CommentTimeDec 16th 2019

    Thanks, Evangelos!

    • CommentRowNumber4.
    • CommentAuthorGuest
    • CommentTimeAug 18th 2021
    Guest comment from Yemon Choi, who can't find his password. I happened to get sent to the page below while searching for some definitions about closed categories, and the claim at

    https://ncatlab.org/nlab/show/tensor+product+of+Banach+spaces

    that Hilb is a closed monoidal category with respect to the Hilbertian tensor product is grievously untrue unless you restrict yourself to finite-dimensional settings and allow yourself non-isometric isomorphisms. Are people actually checking things that they copy into the nLab?

    The book of Ryan that is listed in the references is a good place to learn about tensor products of Banach spaces, provided that it is actually read.
    • CommentRowNumber5.
    • CommentAuthorGuest
    • CommentTimeAug 18th 2021
    BTW the "correct" way to get at the Hilbertian tensor product of two Hilbert spaces is to view the two spaces as "column Hilbert spaces" in the category of **operator spaces**, then take the "projective tensor product" in **that** category, and apply the "underlying Banach space functor". The OS-projective tensor product is the one that makes OpSp into a closed symmetric monoidal category.
    • CommentRowNumber6.
    • CommentAuthorUrs
    • CommentTimeAug 18th 2021

    Why not just fix what needs fixing.

    The line in question seems to be the last one from the Idea-section, introduced in rev 1 from 2012.

    I gather the point is that the adjective “closed” needs to be removed or qualified here.

    Clearly, this must have been written with finite dimensional spaces and linear maps in mind. Which is unfortunate in its lack of qualification, and more so in the context of this page, but has become fairly common habit (e.g. Selinger 2018, slide 21) for authors coming from QM applications. Or it was just a slip of tongue, which is suggested by the use of the word “also” and the fact that closure isn’t claimed the line before.

    In any case, removing that word from the informal Idea-section would seem to fix the problem? If you then have energy left to add your expertise to the entry, as indicated in #5, that would be a great service to the community and would be much appreciated.

    • CommentRowNumber7.
    • CommentAuthorUrs
    • CommentTimeAug 19th 2021

    just to bump this up, re the complaint in #4.

    I have removed that second “closed” from the Idea section, and touched the list of references.

    But I am not invested in this entry and haven’t even really looked into it. Hopefully somebody who cares has the energy to look into it.

    diff, v16, current

    • CommentRowNumber8.
    • CommentAuthorTom Mainiero
    • CommentTimeJun 24th 2022

    I added a remark about Schmidt decomposition working in any dimension along with (shameless self-) reference to a proof in an MO question.

    diff, v17, current

    • CommentRowNumber9.
    • CommentAuthorTom Mainiero
    • CommentTimeJun 24th 2022
    Also, the Schmidt decomposition seems a bit out of place for this page. Just mentioning that if anyone has a better idea of where to move it.

1 to 9 of 9

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