Continuity of linear map on space of distributions

Question

I'm trying to understand how the linear mappings on the spaces of distributions $\mathcal{D}'(\Omega)$ and $\mathcal{E}'(\Omega)$(dist. with compact support in $\Omega$) are characterized ($\Omega$ open subbset of $\mathbb{R}^n$). If I get it right the topologies that one usually uses (the polar topologies?) are generated by semi-norms of the forms (on $\mathcal{D}'(\Omega)$ for example): $p_{\phi, B}(u)=\sup_{\phi \in B } |(u,\phi(x))|$, where $B$ is a bounded subset of $\mathcal{D}(\Omega)$. So if I want to show that a linear map from $T: \mathcal{D}'(\Omega) \to \mathcal{D}'(\Omega)$ to be continuous is it enough to find two semi-norms and a constant such that $p_{\phi, B}(Tu)\leq C p_{\psi, A}(u)$?

Also I wonder about sequential continuity in these spaces; is in general a sequentially continuous linear map $T: \mathcal{D}'(\Omega) \to \mathcal{D}'(\Omega)$ continuous?

I hope this didn't get messy, pleas ask if you want me to clarify anything!

Answer 1

...If I get it right the topologies that one usually uses...

There are two topologies we usually use in $\mathcal{D}'(\Omega)$: the strong topology and the weak star topology.

The "strong topology" is the topology you have described:

• Let $\mathcal{F}$ be the collection of all bounded subsets of $\mathcal{D}(\Omega)$. For each $B\in\mathcal{F}$, the application $p_{B}:\mathcal{D}'(\Omega)\to\mathbb R$ given by $$p_B(u)=\sup_{\varphi\in B}|\langle u,\varphi\rangle|$$ is a semi-norm on $\mathcal{D}'(\Omega)$. The strong topology on $\mathcal{D}'(\Omega)$ is the topology defined by the family $(p_{B})_{B\in\mathcal{F}}$.

This topology is used, for example, by Schwartz (p. 71) and Horvath (p. 313). (Actually, in Horvath's definition $\mathcal{F}$ is replaced by the collection $\mathcal{G}$ of all weak-bounded subsets of $\mathcal{D}(\Omega)$. But this changes nothing because $\mathcal{D}(\Omega)$ is a locally convex Hausdorff space and thus $\mathcal{G}=\mathcal{F}$ (see Rudin, p. 70).)

The "weak* topology" is the usual one:

• The weak star topology on $\mathcal{D}'(\Omega)$ is the weakest topology on $\mathcal{D}'(\Omega)$ that makes every evaluation map $\text{ev}_\varphi:\mathcal{D}'(\Omega)\ni u\mapsto \text{ev}_\varphi(u)=\langle u,\varphi\rangle\in \mathbb K$ continuous.

This topology is used, for example, by Rudin (p. 160) and Grubb (p. 27). (Actually, Grubb uses the equivalent definition via seminorms.)

...is in general a sequentially continuous linear map $T: \mathcal{D}'(\Omega) \to \mathcal{D}'(\Omega)$ continuous?

If $\mathcal{D}'(\Omega)$ is equipped with the weak* topology, the answer is yes. In fact, it is a particular case of the following result, whose proof can be found here.

Let $E$ be a separable complete locally convex space. Let $E'$ be equipped with the weak*-topology. Then, any sequentially continuous linear map $T:E'\to E'$ is continuous.

If $\mathcal{D}'(\Omega)$ is equipped with the strong topology, the answer is also yes (because such a space is bornological (see Horvath, p. 314)). In fact, it is a particular case of the following result.

If $X$ is bornological and $Y$ is locally convex, then any sequentially continuous linear operator $T:X\to Y$ is continuous (Narici & Beckenstein, p. 452).

Remark. The topologies defined above have exactly the same convergent sequences (see p. 74 in Schwartz, or p. 358 in Treves). This explains why, in some contexts (like in the applications to partial differential equations), one usually

• defines only one topology in $\mathcal{D}'(\Omega)$: the simplest one, which is the weak* topology (as Le Dret and Hörmander).

or

• defines no topology in $\mathcal{D}'(\Omega)$ but only the notion of convergence (as Dautray and this other Schwartz's book).