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Can you think of quasi-norms which aren't norms?

  1. Jan 27, 2013 #1
    Just to be clear, a quasi-norm is like a norm but instead of genuine subadditivity we have
    ##||x+y||\leq C(||x||+||y||)## where ##C\geq1## is some fixed constant.

    To be honest, other than trivial examples the only one that comes to mind is ##L^p## for ##p\in(0,1)##. A quick google search doesn't yield much.

    More generally, how about quasi-metric spaces? Similarly the triangle inequality is weakened to have a fixed multiplicative constant out front.
     
  2. jcsd
  3. Jan 28, 2013 #2

    mfb

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    Staff: Mentor

    In finite vector spaces, it is easy to find examples, as the existence of the constant is trivial.

    One weird example for R^2:
    ||x|| = min(|x_1|+2*|x_2|, 2*|x_1|+|x_2|) with some base (e_1,e_2) and C=2.

    Possible generalization: Define an arbitrary function ##f: \{x| x \in V, ||x||=c\} \to R## with ##E>f(x)>e>0## for some e, E. Let ||.||q be a quasi-norm with f(x)=||x||q and use linearity ||ax||q=|a| ||ax||q to extend this to a definition of ||.||q everywhere.
    I did not check this in detail, but it should satisfy all axioms.
     
  4. Jan 31, 2013 #3
    Well, thanks for that. I was hoping for more meaningful examples. I don't work much (really, at all) with Hardy spaces, but apparently ##H^p## for ##p\in(0,1)## is also a quasi-norm space. Probably some Triebel-Lizorkin and Besov spaces are quasi-norm spaces for appropriate parameters.
     
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