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It is known that $C^\infty([0,1])$ is a Frechet space with the seminorms $||f||_k:=||f^{(k)}||_{L^\infty}$.

Question: Are the following spaces still Frechet space? If not, do you know some good counter-examples? $$ C^\infty(\mathbb R), \ \ \ C^\infty_c(\mathbb R), \ \ \ \mathcal S(\mathbb R) $$ where the last one is the Schwarz space.

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    $\begingroup$ You have semi-norms $\|f^{(n)}\|_K$ for any compactum $K$. Since $\Bbb R$ is $\sigma$ compact the topology of these semi-norms is generated by countably many, pass to the metrics $\|f-g_\alpha\|/(1+\|f-g\|_\alpha)$ and sum them up with prefactors $2^{-n}$. $\endgroup$ – s.harp Jul 14 '17 at 23:15
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The spaces $C^\infty(\mathbb R^n)$ and $\mathcal S(\mathbb R^n)$ are Frechet, with respective seminorms $$\|f\|_{l,K} = \sup\limits_{|\alpha| \leq l, x \in K} |\partial^\alpha f(x)|$$ for $l \in \mathbb N$ and $K \subset \mathbb R^n$ compact, and $$\|f\|_{j,k} = \sup\limits_{|\alpha| \leq k, x \in \mathbb R^n} (1 + |x|)^j |\partial^\alpha f(x)|,$$ where $j,k \in \mathbb N$.

The space $\mathcal D(\mathbb R^n) = C_c^\infty(\mathbb R^n)$ is not a Frechet space, and indeed it is not metrizable. Note that $f_k \to f$ in $\mathcal D(\mathbb R^n)$ if there exists a compact set $K$ such that $f_k$ is supported in $K$ for all $k \in \mathbb N$, and if $\partial^\alpha f_k \to \partial^\alpha f$ uniformly for every multi-index $\alpha$. Let us consider $\mathbb R$, and for each $l \in \mathbb N$, let $g_l$ be a non-zero function supported in $[l,l+1]$. Then ${1 \over m} g_l \to 0$ in $\mathcal D(\mathbb R)$ as $m \to \infty$. If $\mathcal D(\mathbb R)$ were metrizable with metric $d$, there would exist $m_l$ large enough that $d({1 \over m_l} g_l,0) \leq {1 \over l}$. Suppose this is the case, and let $f_l = {1 \over m_l} g_l$. We have that $f_l \to 0$ in $\mathcal D(\mathbb R)$. But this cannot be the case, since $\bigcup\limits_{l=1}^\infty \text{supp}(f_l)$ does not have compact closure.

Edit: a little category theoretic language can be clarifying here. A limit of a countable, directed family of Banach spaces is a Frechet space, and this is what $C^\infty(\mathbb R^n)$ and $\mathcal S(\mathbb R^n)$ are. On the other hand, a colimit of even a countable family of Banach spaces is not, in general, even metrizable. Note that $\mathcal D(\mathbb R^n)$ is the colimit of the spaces $C^\infty_K(\mathbb R^n)$ of smooth functions supported in the compact set $K$, with inclusions $C^\infty_K \hookrightarrow C^\infty_{K'}$ when $K \subset K'$. Each space $C^\infty_K$ is itself Frechet.

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