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nLab > Latest Changes: Reflexive Banach space

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    • CommentRowNumber1.
    • CommentAuthorYemon Choi
    • CommentTimeOct 25th 2010

    I started editing the page on reflexive Banach spaces - in particular I corrected the definition and stuck in a mention of "James space". A link or reference is needed but I am currently a bit too frazzled/stressed to do further editing today.

    • CommentRowNumber2.
    • CommentAuthorYemon Choi
    • CommentTimeNov 30th 2011

    Added a remark about closed subspaces of reflexive subspaces being reflexive in non-dream mathematics, while in dream mathematics the reflexive space l ∞l^\infty has the closed non-reflexive subspace c 0c_0.

    • CommentRowNumber3.
    • CommentAuthorTom Leinster
    • CommentTimeNov 30th 2011

    Yemon, does “dream mathematics” mean something more specific than just “what would happen if everything was nice”? I know there are such things as “Mori dream spaces”, for example, and though I’m pretty sure these aren’t what you’re talking about, it is at least precedent for this kind of usage. If you’re using “dream mathematics” to mean something specific, I’d be interested to read about it.

    • CommentRowNumber4.
    • CommentAuthorYemon Choi
    • CommentTimeNov 30th 2011

    Tom, someone else used the term “dream mathematics” on that page before I did. As I am perhaps feeling more intemperate than is productive, I will withhold comment on quite how I feel about that. I merely wished to point out that in dream mathematics, subobjects of reflexives aren’t reflexive. From my perspective, that’s a shame; perhaps from other perspectives it’s a virtue.

    • CommentRowNumber5.
    • CommentAuthorTom Leinster
    • CommentTimeNov 30th 2011

    Ah, I made the classic mistake of assuming that the most recent editor of a page was the author of everything on it. Silly me. Well, I’ll just have to dream about what dream mathematics might be.

    • CommentRowNumber6.
    • CommentAuthorTodd_Trimble
    • CommentTimeNov 30th 2011
    • (edited Nov 30th 2011)

    Toby was the one who introduced the term ’dream mathematics’ on that page. There are a few other instances of that phrase scattered around the nLab; for example, one reads at axiom of choice

    The assumption that every subset of the real line has the Baire property (BP) is consistent with DC but not AC; the same holds for the assumption that every subset of the real line is measurable (LM) if at least one Grothendieck universe exists. These assumptions leads to a very nice setting for analysis called dream mathematics.

    My feeling based on this is that “dream mathematics” has a specific technical meaning and was not simply dreamed up by Toby, but I don’t know. It would be good to create that page and clarify the situation.

    • CommentRowNumber7.
    • CommentAuthorYemon Choi
    • CommentTimeDec 1st 2011
    More edits: added some stuff concerning the dual space functor and the adjunction that arises. (This seemed to fit on this page since one wants to name the canonical embedding of a Banach space into its double dual, and to use its naturality, by which time you might as well notice that it's the unit of an adjunction.)
    • CommentRowNumber8.
    • CommentAuthorTobyBartels
    • CommentTimeDec 4th 2011

    For ‘in dream mathematics’, one might also read ‘in the Solovay model’, if that rings a bell to anybody. I’ve now written dream mathematics.

    • CommentRowNumber9.
    • CommentAuthorTobyBartels
    • CommentTimeDec 4th 2011

    For the latter, there is its own thread.

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