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nLab > Latest Changes: p-norm

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    • CommentRowNumber1.
    • CommentAuthorUrs
    • CommentTimeApr 1st 2017
    • (edited Apr 1st 2017)

    For completeness and for linking-purposes, I thought an entry p-norm was missing. So I created one.

    I have added cross-links with the Idea-sections at sequence space and Lebesgue space and with the Examples-section at normed vector space. Created a stub for Minkowski’s inequality, so far containing essentially nothing but a pointer to Todd’s p-norms (toddtrimble), which I vote for copying to there.

    • CommentRowNumber2.
    • CommentAuthorTodd_Trimble
    • CommentTimeApr 1st 2017

    Thanks, Urs. I didn’t bring that entry on my page to a form I was satisfied with, but I may come back to wrap it up, and then I’d consider copying the contents over to the nLab.

    • CommentRowNumber3.
    • CommentAuthorTobyBartels
    • CommentTimeApr 2nd 2017
    • (edited Apr 2nd 2017)

    I added some material on p<1p \lt 1.

    There is a great deal on Minkowski's inequality already at Lebesgue space, which is where Urs's link above points to (the stub is at Minkowki’s inequality). Perhaps you can stock your stub by moving that material, Urs; if not, then I would put it at p-norm preferably to Lebesgue space.

    • CommentRowNumber4.
    • CommentAuthorUrs
    • CommentTimeApr 2nd 2017

    Toby, thanks for alerting me of the material on Minkowski’s inequality over at Lebesgue space. I had indeed missed that, sorry. Now I have copied the proof given there to Minkowski’s inequality. It’s a bit redundant, but I think it does not hurt to have this duplication.

    Over at Lebesgue space, is there not a lack of emphasis that one needs to divide out functions ff whose ‖f‖ p=0\Vert f \Vert_p = 0 before getting an actual norm?

    • CommentRowNumber5.
    • CommentAuthorTodd_Trimble
    • CommentTimeApr 2nd 2017

    To un-gray a link at Lebesgue space, I’ve written Hölder’s inequality. Now just about everything I was driving at in my web notes is incorporated into the nLab. There’s still some cross-linking to do.

    • CommentRowNumber6.
    • CommentAuthorUrs
    • CommentTimeApr 2nd 2017

    Thanks, Todd!!

    • CommentRowNumber7.
    • CommentAuthorTobyBartels
    • CommentTimeApr 3rd 2017

    @Urs #4:

    Good, I thought that you had rejected that in favour of what Todd had written or something.

    Over at Lebesgue space, is there not a lack of emphasis that one needs to divide out functions ff whose ‖f‖ p=0\Vert f \Vert_p = 0 before getting an actual norm?

    I suppose. The statement does say ‘(equivalence classes of)’ in an appropriate space, but it's rather brief.

    • CommentRowNumber8.
    • CommentAuthorUrs
    • CommentTimeApr 3rd 2017

    Okay, I have made that a tad more explicit here.

    • CommentRowNumber9.
    • CommentAuthorTobyBartels
    • CommentTimeApr 3rd 2017
    • (edited Apr 3rd 2017)

    This might be a good place to remind everybody that, when you use vertical bars (single or double) as delimiters in iTeX, you generally need to enclose them within braces (or use \left and \right) to ensure proper spacing. (This actually applies to any delimiters that are either undirected or op-directed.) Thus,

    $$ |x| - \|x\| - [x[ - ]x] $$

    produces

    |x|−‖x‖−[x[−]x] |x| - \|x\| - [x[ - ]x]

    (which is incorrect), but

    $$ {|x|} - {\|x\|} - {[x[} - {]x]} $$

    produces

    |x|−‖x‖−[x[−]x] {|x|} - {\|x\|} - {[x[} - {]x]}

    (which is correct), and

    $$ \left|x\right| - \left\|x\right\| $$

    produces

    |x|−‖x‖ \left|x\right| - \left\|x\right\|

    (which is also correct, but \left[x\right[ produces an error in iTeX).

    Note that iTeX does not support \lvert, \rvert, \lVert, and \rVert, which are directed versions of | and \| (although it supports the synonyms \vert and \Vert for the undirected versions), nor does it support \mathopen and \mathclose (which is how you’re supposed to apply the proper directions to undirected and op-directed delimiters in real TeX).

    In some cases, you need braces inside the delimiters too! Thus, even

    $$ {|-|} - {\|-\|} - {[-[} - {]-]} $$

    produces

    |−|−‖−‖−[−[−]−] \left] {|-|} - {\|-\|} - {[-[} - {]-]}

    (which is incorrect), but

    $$ {|{-}|} - {\|{-}\|} - {[{-}[} - {]{-}]} $$

    produces

    |−|−‖−‖−[−[−]−] {|{-}|} - {\|{-}\|} - {[{-}[} - {]{-}]}

    (which is correct), while

    $$ \left|-\right| - \left\|-\right\| $$

    produces

    |−|−‖−‖ \left|-\right| - \left\|-\right\|

    (which is correct, as is using \lvert etc or \mathopen and \mathclose in real TeX.)

    I usually put in the outer braces as a matter of good habit, even though they may not be necessary (depending on the surrounding characters). I usually don't bother with the inner braces, since they are needed so rarely, but I have used them in the past.

    In principle, superscripts and subscripts should be inside the braces to get the correct heights, although it seems to me that this won't make a visible difference unless your delimiters are too small anyway. Thus,

    $$ {|\frac{x}{y}|}^2 - {\Big|\frac{x}{y}\Big|}^2 - {\left|\frac{x}{y}\right|}^2 $$

    produces

    |xy| 2−|xy| 2−|xy| 2 {|\frac{x}{y}|}^2 - {\Big|\frac{x}{y}\Big|}^2 - {\left|\frac{x}{y}\right|}^2

    (which is in principle incorrect), and

    $$ {|\frac{x}{y}|^2} - {\Big|\frac{x}{y}\Big|^2} - {\left|\frac{x}{y}\right|^2} $$

    produces

    |xy| 2−|xy| 2−|xy| 2 {|\frac{x}{y}|^2} - {\Big|\frac{x}{y}\Big|^2} - {\left|\frac{x}{y}\right|^2}

    (which is in principle correct, although I can only tell the difference in the first term, which one should not really use).

    • CommentRowNumber10.
    • CommentAuthorTobyBartels
    • CommentTimeApr 3rd 2017

    (If you read the previous comment before this comment appeared, then look at it again for additions.)

    • CommentRowNumber11.
    • CommentAuthorUrs
    • CommentTimeApr 4th 2017
    • (edited Apr 4th 2017)

    Thanks, yes, I am aware of this and I keep adding these extra braces in my edits. Did I not in this case? Sorry.

    • CommentRowNumber12.
    • CommentAuthorTodd_Trimble
    • CommentTimeApr 4th 2017

    Toby might have also been tacitly addressing me (I also knew it but hadn’t followed through in all cases). In any case, I think I’ve fixed just about all of them.

    • CommentRowNumber13.
    • CommentAuthorTobyBartels
    • CommentTimeApr 4th 2017

    My comment #9 is a general one, not addressed to anybody in particular. (I'm pretty sure that Urs is adding these braces most of the time, if not always, but I also know that I forget them occasionally.) Since I had to fix some, I put in the reminder, without worrying about who it was that left them out.

    • CommentRowNumber14.
    • CommentAuthorUrs
    • CommentTimeApr 4th 2017
    • (edited Apr 4th 2017)

    You should move your #9 to the HowTo so that next time you can send a reminder just by giving a pointer!

    • CommentRowNumber15.
    • CommentAuthorTobyBartels
    • CommentTimeApr 4th 2017

    OK, I wrote HowTo#vbar.

    • CommentRowNumber16.
    • CommentAuthorUrs
    • CommentTimeApr 4th 2017

    Thanks!

    • CommentRowNumber17.
    • CommentAuthorUrs
    • CommentTimeApr 17th 2017

    At p-norm I moved the technical discussion of generalizations for 0≤p<10 \leq p \lt 1 out of the Idea-section, into a dedicated new subsection Generalizations.

    • CommentRowNumber18.
    • CommentAuthorTobyBartels
    • CommentTimeApr 17th 2017

    Good idea, and it allows one to state the facts about these a little more leisurely.

  19. added example of p=1p = 1 which is the taxicab or Manhattan norm

    Anonymous

    diff, v10, current

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