# Understanding the weak norm and it's notation

1. Apr 5, 2012

### PhillipKP

Hello.

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

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

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 $\Big\{i \,|\, |x_i| > \epsilon \Big\}$ 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. Apr 5, 2012

### micromass

It is the set of all elements i such that $|x_i|>\varepsilon$.

For example, if $\varepsilon=2$ and

$$x=(2,5,3,1,2,6)$$

Then the second, third and sixth element are $>\varepsilon$.
So $\{i~\vert~|x_i|>2\}=\{2,3,6\}$.

3. Apr 5, 2012

### PhillipKP

Thank you! That was nice and quick!

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