# Zero “norm” properties

I have seen the claim that the l0-norm ($\|X\|_0$ = support(X)) is a pseudo-norm because it does not satisfy all properties of a norm. I thought it to be triangle inequality, but am not able to show it by example. Can anyone give an example to show that the l0-norm does not satisfy the triangle inequality?

Thanks.

• It does not violate triangle inequality. It's not positive homogeneous. – S.B. Jun 21 '13 at 4:03
• This can't be the support of $X$. It must be its measure for some measure. The counting measure if $X$ is a finite-dimensional vector. The measure at use if $X$ is some measurable function. – Julien Jun 21 '13 at 15:03

This is a misuse of the term seminorm, which is defined as a norm, except that it's okay for a nonzero element to have seminorm $0$.
But the property $\|\lambda x\|=|\lambda|\|x\|$ clearly fails for $l_0$: instead we have $\|\lambda x\|_0=\|x\|_0$ for all $\lambda\ne 0$.