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

  • Thread starter danzibr
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In summary, a quasi-norm is a weakened version of a norm, where a fixed constant is added to the subadditivity requirement. Examples include ##L^p## for ##p\in(0,1)##, Hardy spaces, and Triebel-Lizorkin and Besov spaces. In finite vector spaces, finding examples is trivial, but in general spaces, more work is needed to find meaningful examples.
  • #1
danzibr
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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.
 
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  • #2
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.
 
  • #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.
 

1. Can you explain the difference between norms and quasi-norms?

Norms are mathematical functions that measure the size or magnitude of a vector. They satisfy certain properties such as non-negativity, homogeneity, and the triangle inequality. Quasi-norms, on the other hand, may not satisfy all of these properties, making them weaker than norms.

2. How can quasi-norms be useful in scientific research?

Quasi-norms can be used in situations where it is not necessary to have all of the properties of a norm, but some level of size or magnitude comparison is still needed. They can also be used as a stepping stone in developing new mathematical theories and models.

3. Are there any real-life examples of quasi-norms?

Yes, there are many examples of quasi-norms in various fields. In economics, the Gini coefficient is a quasi-norm used to measure income inequality. In physics, the Minkowski distance is a quasi-norm used to measure the separation between two events in space-time.

4. Can a quasi-norm ever be equal to a norm?

Yes, there are cases where a quasi-norm can be equal to a norm. This can happen when all of the properties of a norm are satisfied, making the quasi-norm a fully functional norm. However, this is not always the case, as quasi-norms are generally weaker than norms.

5. How are quasi-norms related to other mathematical concepts?

Quasi-norms are closely related to other mathematical concepts such as metrics and semi-norms. Metrics are functions that measure the distance between two points, while semi-norms are similar to norms but may not satisfy the triangle inequality. Quasi-norms can be seen as a combination of these two concepts, incorporating some properties of metrics and some properties of norms.

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