View Single Post

Mentor
 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 re-summarize?
OK, let's focus on "distributions" first, not "tempered distributions". We define the test function space (I'll call it $\mathcal D$) as the set of $C^\infty$ functions from $\mathbb R^n$ into $\mathbb C$ that have compact support. I would like to define a "distribution" as a member of the dual space $\mathcal D^*$, defined as the set of continuous linear functions $T:\mathcal D\rightarrow\mathbb C$. 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.

$T:\mathcal D\rightarrow\mathbb C$ is continuous at $g\in\mathcal D$ if for each $\epsilon>0$ there's a $\delta>0$ such that $\|f-g\|<\delta\implies|T(f)-T(g)|<\epsilon$. T is continous on a set $U\subset\mathcal D$ if it's continuous at each point in U. Alternatively, and equivalently, T is continuous on $U\subset\mathcal D$ if $T^{-1}(E)$ is open for every open $E\subset\mathbb C$.

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.