Quote by strangerep
I'm starting to lose track of what you're referring to, hence unable to offer a pinpoint
answer. Maybe you should resummarize?

OK, let's focus on "distributions" first, not "tempered distributions". We define the test function space (I'll call it [itex]\mathcal D[/itex]) as the set of [itex]C^\infty[/itex] functions from [itex]\mathbb R^n[/itex] into [itex]\mathbb C[/itex] that have compact support. I would like to define a "distribution" as a member of the dual space [itex]\mathcal D^*[/itex], defined as the set of continuous linear functions [itex]T:\mathcal D\rightarrow\mathbb C[/itex]. But to do that, we need to define what "continuous" means. There are at least two ways to do that.
Option 1: Define the usual inner product. The inner product gives us a norm, and the norm gives us a metric. Now we can use the definition of continuity that applies to all metric spaces.
[itex]T:\mathcal D\rightarrow\mathbb C[/itex] is continuous at [itex]g\in\mathcal D[/itex] if for each [itex]\epsilon>0[/itex] there's a [itex]\delta>0[/itex] such that [itex]\fg\<\delta\impliesT(f)T(g)<\epsilon[/itex]. T is continous on a set [itex]U\subset\mathcal D[/itex] if it's continuous at each point in U. Alternatively, and equivalently, T is continuous on [itex]U\subset\mathcal D[/itex] if [itex]T^{1}(E)[/itex] is open for every open [itex]E\subset\mathbb C[/itex].
Weird, it takes a lot less to cause a database error now than a couple of weeks ago. I'll continue in the next post.