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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.

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