There are several questions+answers on this site demonstrating that the topology of $\mathcal D(\mathbb R)$, the space of smooth compactly supported test functions on $\mathbb R$, is not Frechet (and indeed not even sequential). However, they deal with the first definition of Frechet spaces on Wikipedia, whereas I am more used to using the second definition in terms of a countable family of seminorms. Specifically, I want to understand which of the three criteria in the second Wikipedia definition fail for $\mathcal D ( \mathbb R)$. Why can't we use $\|\phi\|_k = \|D_k \phi\|_\infty$ as the relevant family of seminorms? And isn't this family complete, provided that all test functions are supported in the same compact interval?
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$\begingroup$ The family of seminorms $\phi{K, n}(f)=\max_{1\leq j\leq n}\{\sup_{x\in K}|D^jf(x)|\}$ for each $K$ compact is complete in each $K$. the issue is the glue all those seminorms for different compact sets that cover $\mathbb{R}$. The inductive topology procedure does that. See for example Rudin's FA book, chapter 6. $\endgroup$– MittensCommented Oct 15, 2022 at 15:29
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The $\|D_k\phi\|_{\infty}$ do not define the topology of $\mathcal{D}(\mathbb{R})$. To see an "explicit" and obviously non-countable collection of seminorms which do define the topology of $\mathcal{D}(\mathbb{R})$, look at Example 3 in my answer
answered Jan 6, 2023 at 1:02