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

[danzibr]

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.

 

[mfb]

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.

 

[danzibr]

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.