This comes from W. Rudin's Functional Analysis Exercise 6.6(a). There are some questions on the same topic but none of them have this problem solved at all. For example this and this.
Here are the following questions (b) and (c). I'm not asking about these two questions here but they may help us understand what happens for (a):
(b) Let $\Omega$ be open in $R^n$. Suppose $\Lambda_i \in \mathscr{D}(\Omega)$, and suppose that all $\Lambda_i$ have their supports in some fixed compact $K \subset \Omega$. Prove that the sequence $\{\Lambda_i\}$ cannot converge in $\mathscr{D}'(\Omega)$ unless the orders of the $\Lambda_j$ are bounded.
It is followed by (c):
(c) Can the assumption about the supports be dropped in (b)?
So it's likely that (a) serves as a counterexample.
In case you are confused with the notations, I'd like to elaborate.
- $D$ of course means differentiation.
- $\mathscr{D}(\Omega)$ is the test function space and $\mathscr{D}'(\Omega)$ is the distribution space, i.e. the dual of $\mathscr{D}(\Omega)$.
- $\Lambda \in \mathscr{D}'(\Omega)$ if and only if, to every compact $K \subset \Omega$, there corresponds a nonnegative integer $N$ and a constant $C<\infty$ such that
$$ |\Lambda\phi| \le C \| \phi\|_N $$
for every smooth function with support in $K$. Here
$$ \| \phi \|_N = \max \{ |D^\alpha \phi|:x \in \Omega,|\alpha| \le N \} $$
If $\Lambda$ is such that one $N$ will do for all $K$ (but $C$ is not necessarily the same), then the smallest such $N$ is called the order of $\Lambda$. In particular, if $\Lambda$ has compact support, then the order is finite.
How should I study this argument? I think it is pretty unrealistic to construct a function explicitly. I'm thinking about the linear functional $\Gamma_n(\phi)=\sum_{k=1}^{n}\exp\{-(n!)!\}D^n\phi(0)$, but how can we make use of Banach-Steinhaus theorem, for example? What about the order and support of this functional?
Another question. Should we consider smooth $\phi$ with compact support first, since there we are talking about distribution theory in particular?
Edit 1: I'm not asking about (b) and (c), I added them for the sake of context. Hope confusion is prevented.