$C_0^\infty(R)$ metrizable? I found the statement that the space $C_0^\infty(\mathbb{R})$ is not metrizable:
see for example Remark 27 here:
http://www.math.mcgill.ca/gantumur/math581w14/distrib.pdf
or one of the answers here:
https://mathoverflow.net/questions/52032/examples-of-non-metrizable-spaces
On the other hand, $\|f\|:=sup_{x\in\mathbb{R}} |f(x)|$ defines a norm on this space and therefore the space is a metric space. But, as far as I know, every metric space is metrizable which is a contradiction to the statement above that the space of continuous and compactly supported functions is not metrizable.
Am I misunderstanding something? Can anyone help me?
Thank you very much!
 A: What you are missing is the context in which the statement is being made. Notice that the notes you linked to your question are about Distributions. In the later link, it is implicit that they are referring to "test" functions in the theory of distributions
In the theory of distributions, it is understood (certainly not for those who study this for the first time) that $\mathcal{C}^\infty_0$, which is then denoted as $\mathcal{D}$ and called test functions, is assigned an inductive limit topology -not easy to digest the first time arround- that make it a locally convex complete linear space.
Although the topology is not metrizable, its raison d'etre is to justify the nice properties of its dual (the space of distributions or generalized functions), some of which which were in used in Physics long before they were rigorously justified. Distributions (and its cousin, tempered distributions), it turns out,  is a  great tool (or the right tool) to solve and analyze  some partial differential equations (PDEs).
