I really don't understand why the following is a seminorm rather than a norm? $$ p_k(u)=\sum_{|α|\le k}\sup_{x∈R^n}(1+|x|^2)^{k/2}|D^α u(x)|, $$ for all $u \in C^\infty$.
I do understand if $|α|>0$, then $p_k(u)$ has to be a seminorm because of the example $u(x)=$constant.
(This staff is about tempered distributions. See Michael Taylor's book "Partial differential equations I").
I can't figure it out. I has been blocked by this for a while and I don't know what I am missing. Hoping for your answers, thanks.
$p_k$ is only real valued for functions $u\in\mathscr S$ (Schwartz' space of rapidly decreasing smooth functions). However, restricted to that space $p_k$ is indeed a norm since for all $x\in \mathbb R^n$ you have $$|u(x)|\le (1+|x|^2)^{k/2} |D^0u(x)| \le p_k(u).$$
The reason for calling it nevertheless a seminorm is probably that the general theory of locally convex spaces in made with seminorms.