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    • CommentRowNumber1.
    • CommentAuthorTobyBartels
    • CommentTimeAug 17th 2011

    I added some more to Lebesgue space about the cases where 1<p<1 \lt p \lt \infty fails.

    • CommentRowNumber2.
    • CommentAuthorTodd_Trimble
    • CommentTimeAug 17th 2011
    • (edited Aug 17th 2011)

    Hi Toby. Could you check again the asserted local convexity in the cases 0<p<10 \lt p \lt 1? Because the Hahn-Banach theorem implies that the dual of a locally convex TVS is non-zero, whereas it is known that L pL^p for this range of pp typically has zero dual. See also this section from Wikipedia.

    • CommentRowNumber3.
    • CommentAuthorTobyBartels
    • CommentTimeAug 18th 2011

    Yeah, I remembered that wrong. Fixed.

    • CommentRowNumber4.
    • CommentAuthorTodd_Trimble
    • CommentTimeAug 18th 2011
    • (edited Aug 18th 2011)

    I added a section on Minkowski’s inequality for the case 1p1 \leq p \leq \infty, with a proof of my own devising. I don’t think it’s actually original with me, but I’ve not seen it in the books I’ve looked at. The textbook proofs I have seen involve Hölder’s inequality, but without the courtesy of saying what is going on in that proof conceptually. I have a page on these issues on my lab, here.

    • CommentRowNumber5.
    • CommentAuthorTobyBartels
    • CommentTimeAug 18th 2011

    Cool!

    By the way, when putting norms and absolute values in itex, it looks a lot better if you put each one inside braces. Compare:

    • |x| = |y| produces ‘|x|=|y||x| = |y|’;
    • {|x|} = {|y|} produces ‘|x|=|y|{|x|} = {|y|}’.
    • CommentRowNumber6.
    • CommentAuthorTodd_Trimble
    • CommentTimeAug 18th 2011

    Thanks for the tip! “Who knew?”

    • CommentRowNumber7.
    • CommentAuthorTobyBartels
    • CommentTimeAug 18th 2011

    Now that I’m looking at this on my phone, they look identical (and, unusually, better). That’s weird!

    • CommentRowNumber8.
    • CommentAuthorzskoda
    • CommentTimeAug 18th 2011

    My memory is that between 00 and 11 we have still Frechet spaces.

    • CommentRowNumber9.
    • CommentAuthorTobyBartels
    • CommentTimeAug 18th 2011
    • (edited Aug 18th 2011)

    But Fréchet spaces are locally convex, giving Todd’s objection again. (I seem to recall that FF-spaces have sometimes been called “Fréchet spaces”.)

    • CommentRowNumber10.
    • CommentAuthorTodd_Trimble
    • CommentTimeAug 18th 2011
    • (edited Aug 18th 2011)

    Toby #9 is right. There is some terrible terminological confusion here. I grew up with the meaning of Fréchet space as including local convexity (which rules out L 1/2L^{1/2}), but apparently some people use it to mean what is called F-space.

    A Fréchet space in my meaning is a TVS whose topology is given by a countable family of seminorms under which the TVS becomes a complete metric space (including the axiom that d(x,y)=0d(x, y) = 0 implies x=yx = y). If you have a Fréchet space given by a single seminorm, then that is a norm and you get a Banach space.

    • CommentRowNumber11.
    • CommentAuthorTobyBartels
    • CommentTimeAug 18th 2011

    I clarified that L 0L^0 is not even an FF-space.

    • CommentRowNumber12.
    • CommentAuthorDmitri Pavlov
    • CommentTimeMay 27th 2022

    Added:

    Notation

    For historical reasons (starting with the original paper by Riesz), the exponent pp is traditionally taken to be the reciprocal of the “correct” exponent.

    If we take M p=L 1/pM^p=L^{1/p}, the spaces M pM^p form a C\mathbf{C}-graded algebra, where C\mathbf{C} denotes complex numbers.

    This is a conceptual explanation for the appearance of formulas like 1/p+1/q=1/r1/p+1/q=1/r in Hölder’s inequality.

    In differential geometry, the notion of density does use the “correct” grading.

    In the Tomita–Takesaki theory, the parameter tt for modular automorphism group is almost the “correct” grading, except that it is multiplied by the imaginary unit ii.

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