I'm trying to solve this question with the following given information; I have $\Omega \in \mathbb{R}$ open set, $C^\infty_o(\Omega)$ is a vector space with usual addition and multiplication, and the space $\mathbb{D}(\Omega)$ equipped with the following convergence is the space of test functions.
define the Convergence $\phi_j \rightarrow \phi$ in $\mathbb{D}(\Omega)$ if:
- There exist $K \subset \Omega$ compact such that support$(\phi_j)\subset K,\forall j$.
- $\forall \alpha$ multi-index; $\partial^\alpha \phi_j \rightarrow \partial^\alpha \phi$ uniformly on $\Omega$.
$\mathbb{D}^\prime(\Omega)$ is the set of distributions on $\mathbb{D}(\Omega)$
The question is; if $\Omega = (0,2)$ & $u:\mathbb{D}(\Omega) \rightarrow \mathbb{R} , u(\phi) := \sum_{j=1}^\infty \phi^{(j)}(\frac{1}{j})$ where $(j)$ is the $j^{th}$ derivative of $\phi$.
- (a)Show that $u \in \mathbb{D}^\prime(0,2)$.
- (b)Does u have finite order?
Solution: (a) apparently $u$ is linear and let K$\subset$ (0,2) compact then there exist $N\in \mathbb{Z}^+$, $i.e \frac{1}{N} \in K$, but $\frac{1}{j} \notin K$ for $j>N$ If support$(\phi) \subset K$ then $|u(\phi)| = |\sum_{j=1}^N \phi^{(j)}(\frac{1}{j})| \leq \sum_{j=1}^N max|\phi^{(j)}|$ then I can conclude it is sequentially continuous then it is a distribution and $u \in \mathbb{D}^\prime(0,2)$
My problem with (b) part, I can see from answer in (a) that I can't assume the finite order because of dependence of $N$ on the set $K$ but also I don't know how to prove it is not finite?
Can anyone help?Thank you.
