Let
- $d\in\mathbb N$
- $\Lambda\subseteq\mathbb R^d$ be open
- $k\in\mathbb N_0$
Now, let $$\left\|f\right\|_{C^k(K)}:=\sup_{x\in K}\frac{|f(x)|}{1+|x|}+\sum_{1\le|\alpha|\le k}\sup_{x\in K}|{\rm D}^\alpha f(x)|\;\;\;\text{for }f\in C^k(\Lambda)$$ for compact $K\subseteq\Lambda$. I've seen that $C^k(\Lambda)$ is endowed with the topology generated by the family $$\left\{\left\|\;\cdot\;\right\|_{C^k(K)}:K\subseteq\Lambda\text{ is compact }\right\}$$ of seminorms on $C^k(\Lambda)$.
It might be a stupid question, but what's the reason to use $\sup_{x\in K}\frac{|f(x)|}{1+|x|}$ instead of $\sup_{x\in K}|f(x)|$? Are there any major differences in the generated topological spaces?
Having a decaying weight may be convenient for some estimates, but the topology is the same. Since each $K$ is compact, the expression $1+|x|$ is bounded between two positive constants: $1$ and some constant $C_K$.
Generally, suppose we have two families of seminorms $\{p_\alpha : \alpha \in I\}$ and $\{q_\alpha : \alpha \in I\}$ such that for each $\alpha\in I$ $$ C_\alpha^{-1} p_\alpha \le q_\alpha \le C_\alpha p_\alpha $$ for some constant $C_\alpha$. Then both families induce the same topology.
Indeed, a neighborhood of $0$ in the first topology contains a set of the form $$ \{x : p_{\alpha_i}(x) < r_i,\ i = 1, \dots, n\} $$ which contains $$ \{x : q_{\alpha_i}(x) < r_i/C_\alpha,\ i = 1, \dots, n\} $$ which is a neighborhood of $0$ in the second topology. And conversely.