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.
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 $0$.
The cardinality of support satisfies the triangle inequality, since the support of the sum of two vectors is contained in the union of their supports.
But the property $\|\lambda x\|=|\lambda|\|x\|$ clearly fails for $l_0$: instead we have $\|\lambda x\|_0=\|x\|_0$ for all $\lambda\ne 0$.