Why $\lVert \phi \rVert_{k,m}=max\{\lvert D^{\alpha}\phi (x)\rvert : \lvert \alpha \rvert\le k , x \in K_{m+1}\setminus K_m\}$ just a semi-norm?

Question

Let $\Omega\subset \mathbb{R}^n$ be an open set, $\phi\in C^{\infty}_c$ (i.e. a smooth function with compact support) and $(K_m)_m$ be a sequence of compact sets $K_m\nearrow \Omega$. I was asked to prove that $\lVert \phi \rVert_{k,m}=max\{\lvert D^{\alpha}\phi (x)\rvert : \lvert \alpha \rvert\le k , x \in K_{m+1}\setminus K_m\}$ is a semi-norm. This I can show. But why is it only a semi-norm? What would be an example of a $\phi$ smooth with compact support such that it is different that 0 but it's norm is equal to 0?

Answer 1

Take a bump function $0\neq \varphi$ such that $\operatorname{supp} \varphi \subset K_{m}$.