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I have been wondering about the following concerning the spaces $\mathcal D$ of test functions (say on $\mathbf R$).

  1. It is my understanding that the topology on this space is inductive limit topology. Are there other useful functions spaces like this (i.e. not contrived) whose topology is an inductive limit?

  2. I also know that $\mathcal D$ is not metrizable. Is it first countable? Are they completely regular? Is there a reference with an outline of all the topological properties of these spaces?

  3. Is there a way to define a uniformity on $\mathcal D$ to make them into a uniform space such that the topology it induces coincides with its inductive limit topology?

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3. Yes, every topological vector space (or every abelian topological group for that matter) has a canonical uniformity, in particular $\mathcal{D}$ is completely regular. See e.g. Schaefer's or Kelley's books on topological vector spaces for these general facts. 2. No. A first countable topological vector space is metrizable. –  t.b. Apr 18 '12 at 18:52
    
Excellent. Thanks. –  echoone Apr 19 '12 at 4:23

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