Convergence in $D'$

Question

I was given by my professor of mathematical methods for physicist, a notion of convergence in $D'(\Omega)$, the space of distributions on $D(\Omega)$. Namely:

A sequence of distributions $T_n\in\ D'(\Omega)$ is said to converge to a distribution in $D'(\Omega)$ if $\forall \phi \in D(\Omega)$ $<T_n,\phi>\rightarrow<T,\phi>$.

My question is: in what ways is this notion of convergence meaningful? Is it induced by a norm? If not, why have we chosen this particular notion of convergence?

Is it because it makes the space "complete"? What does it even mean, without a metric (if there isn't one, obviously), to be complete?

Thank you in advance.

Answer 1

Yes, $D'(\Omega)$ is complete but not induced by a norm, basically you have a bunch of seminorms; it is (what is known) as a LF space. A more conceptual viewpoint on the topology on $D'(\Omega)$ is that it is a certain locally convex direct limit of spaces $C^\infty_0(\Omega')$. A good reference for the general framework is Bourbaki's espaces vectoriels topologiques, chapitre II, §4. espaces localement convexes; or Simon/Reed, functional analysis, chapter 4.

Answer 2

First of all, the wiki article on distributions is a good source to start with.

This type of convergence is called the weak-* convergence - more on it here.

The space of distributions is complete under this notion of convergence, but this space is not normed. It is a locally convex space. It is a multinormed space.

For a topological space to be complete means that every Cauchy net (generalisation of Cauchy sequence) has a limit.