# Definition of distribution

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:

1. There exist $$K \subset \Omega$$ compact such that support$$(\phi_j)\subset K,\forall j$$.
2. $$\forall \alpha$$ 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.

Suppose it was finite order, say order $$N$$. Let $$\phi$$ be supported on $$U\subset (0,2)$$ where $$U\cap \{1,\frac12,\frac13,\dots\} = \{\frac1{N+1}\}$$ Then $$u(\phi) = \phi^{(N+1)}(\frac1{N+1})$$...