# On support of distributions in ${\cal D'}(\Omega)$.

First some definitions. $$\Omega \subseteq \mathbb R^N$$.

1. $${\cal D}(\Omega)$$ denotes the set of all compactly supported smooth functions, $$f \in {\cal C}^\infty_0(\Omega)$$, equipped with the following convergence notion:

$${\cal C}^\infty_0(\Omega) \ni \{ f_n \}_{n \in \mathbb N} \overset {n \to \infty}{\longrightarrow} f \in {\cal C}^\infty_0(\Omega)$$ if:

• $$\exists K \subset \Omega$$ compact $$\ |$$ $${\rm supp}(f_n) \subseteq K \quad \forall n$$
• $$\partial^\alpha f_n \rightrightarrows \partial^\alpha f$$ uniformly, $$\forall \alpha$$ multiindex.
1. a linear functional $$u: {\cal D}(\Omega) \to \mathbb C$$ is a distribution if $$\forall$$ convergent sequence $$\{f_n\}_{n \in \mathbb N} \in {\cal D}(\Omega)$$ one has: $$\lim_n u(f_n) = u (\lim_n f_n)$$

$${\cal D}' (\Omega) \equiv \{ u: {\cal D}(\Omega) \to \mathbb C \ | \ u$$ is a distribution$$\}$$

1. given $$\Omega_0 \subset \Omega$$, $$\Omega_0$$ is said to be a vanishing open set for $$u \in {\cal D}'(\Omega)$$ if $$\forall f \in {\cal D}(\Omega)$$ such that $${\rm supp} f \subseteq \Omega_0$$, one has: $$u(f) =0$$

4.$$\forall u \in {\cal D}'(\Omega)$$ we define $${\rm supp } \ u \equiv \Omega \setminus \bigcup\limits^{}_{i} \Omega_0^i$$, with $$\Omega_0^i$$ a vanishing open set for $$u$$.

Question: is $${\rm supp} \ u$$ compact $$\forall u \in {\cal D}'(\Omega)$$?

• Since $\Omega \subseteq \mathbb R^N$, which is finite-dimensional, maybe we could apply Heine-Borel theorem: $\Omega$ is compact $\iff$ $\Omega$ is closed and bounded. It's closed by definition, as a complementary set of a union of open sets which is open, but I'm not sure about boundedness. Jun 27, 2021 at 14:28

No, there are distributions with non-compact support. An example is a pulse train: $$u = \sum_{k=-\infty}^{\infty} \delta_k,$$ where $$\delta_k$$ is a Dirac delta supported at $$x=k$$. Less exotic examples come from non-compactly supported functions in $$L^1_{loc}$$.
• Is ${\rm supp } \ u \equiv \mathbb Z$ in this case? Jun 27, 2021 at 14:41
• Another example is $u(f):=\int_{-\infty}^{\infty} f(x) \, dx.$ It has all of $\mathbb{R}$ as support. Jun 27, 2021 at 17:03