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Understanding the weak norm and it's notation

  1. Apr 5, 2012 #1
    Hello.

    I'm trying to grasp the notation for the definition of something called the weak q-norm, defined as

    [itex]\|x\|_{q,w}^q = \sup\limits_{\epsilon > 0} \epsilon^q \left| \Big\{i \,|\, |x_i| > \epsilon \Big\} \right|[/itex]

    I don't come from a pure math background so I've never seen this notation before but I was wondering if someone knew what the notation inside the "braces" mean:

    Specifically what does [itex] \Big\{i \,|\, |x_i| > \epsilon \Big\} [/itex] mean?

    I found it in the context of a lecture for a class in compressive sensing:
    http://theproofisinthepudding.wordpress.com/2012/01/12/lecture-2/#more-543

    Thanks for any help you can provide.
     
  2. jcsd
  3. Apr 5, 2012 #2

    micromass

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    It is the set of all elements i such that [itex]|x_i|>\varepsilon[/itex].

    For example, if [itex]\varepsilon=2[/itex] and

    [tex]x=(2,5,3,1,2,6)[/tex]

    Then the second, third and sixth element are [itex]>\varepsilon[/itex].
    So [itex]\{i~\vert~|x_i|>2\}=\{2,3,6\}[/itex].
     
  4. Apr 5, 2012 #3
    Thank you! That was nice and quick!
     
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