I want to confirms some questions that pop up when I read my Functional Analysis book by Rudin. Let $K\subset \mathbb{R}^n$ be compact, define $\mathcal{D}_K=\{f\in C^\infty(\mathbb{R}^n):supp(f)\subset K\}$, with seminorms $||f||_N:=\max \{|D^\alpha f(x)|:x\in K,|\alpha|\leq N\}$. I want to confirm whether this "max" always exist? By mean, is there $f\in\mathcal{D}_K$ such that $||f||_N=\infty$?
Let $\Omega\subset\mathbb{R}^n$ be open, define $$\mathcal{D}(\Omega):=\bigcup_{K\subset\Omega\text{ compact}}\mathcal{D}_K$$ called test function space.
I know that $\mathcal{D}_K\subset\mathcal{D}(\Omega)$ is closed. I have a hunch that it is also BOUNDED subset by I have difficulties in showing that. Any idea/hints? Or if it is not true, is there a counter example? Any help is appreciated.
First notice that the set of multi-indices $\alpha$ with $|\alpha|\leq N$ is finite. Therefore it suffices to show that $\sup_{x\in K}|D^\alpha f(x)|<\infty$ for all multi-indices $\alpha$. But this follows simply from the fact that $D^\alpha f$ is continuous by definition and continuous functions are bounded on compact sets.
As is noted on Wikipedia, a non null subspace of a Hausdorff topological vector space is not bounded. However, bounded subsets of $\mathcal{D}_K$ are bounded in $\mathcal{D}(\Omega)$ (this follows from the fact that the topology on $\mathcal{D}(\Omega)$ is generated by all seminorms whose restrictions to all $\mathcal{D}_K$ are continuous).