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The space $C^\infty(\Bbb R)$ is usually defined to be the vector space of smooth functions on $\Bbb R$. We might define $C_0^\infty(\Bbb R)$ as the space of smooth functions whose derivatives all vanish at infinity.

On $C_0^\infty(\Bbb R)$ one has a bunch of semi-norms, namely: $\|f\|_n:=\sup_{x\in\Bbb R}|f^{(n)}(x)|$. These induce a topology and also a uniform structure on $C_0^\infty(\Bbb R)$.

I have a sort of pre-question: This topology/uniform structure does not seem to be normable, is it the standard topology/uniform structure one assigns the this vector space?

My actual question is:

Is $C^\infty_0(\Bbb R)$ complete with this uniform structure?

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I don't know for sure - I haven't seen $C_0^{\infty}(\mathbb{R})$ used much - but I would expect that this is the standard topology/uniform structure on that space. It's rather natural, and has good properties, in particular this family of (semi-)norms makes it a Fréchet space.

We know that $C_0(\mathbb{R})$ endowed with the maximum norm is a Banach space. Hence the product

$$X = C_0(\mathbb{R})^{\mathbb{N}}$$

is a Fréchet space, and we have an embedding

$$\iota \colon C_0^{\infty}(\mathbb{R}) \to X;\quad f \mapsto (f, f', f'', \dotsc).$$

Now one checks that $\iota\bigl(C_0^{\infty}(\mathbb{R})\bigr)$ is a closed subspace of $X$: For $x < y$ and $m \in \mathbb{N}$, the functional

$$\eta_{x,y,m} \colon (g_n) \mapsto g_m(y) - g_m(x) - \int_x^y g_{m+1}(t)\,dt$$

is continuous, hence its kernel is closed, and

$$\iota\bigl(C_0^{\infty}(\mathbb{R})\bigr) = \bigcap_{x,y,m} \ker \eta_{x,y,m}.$$

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    What would be the most natural space to use when talking about smooth functions? For example we don't use $C(\Bbb R)$ over $C_0$ or $C_b$ because it cannot be given a good norm. Thats why I thought $C_0^\infty$ would be the most natural vector space for smooth functions. – 2017-02-05
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    There is no single most natural space. $C(\mathbb{R})$ is a perfectly good Fréchet space. $C^{\infty}(\mathbb{R})$ also, and that one is in fact even better, it's a Fréchet-Montel space, every bounded subset of $C^{\infty}(\mathbb{R})$ is relatively compact. For other things, one uses the space of rapidly decreasing functions, aka Schwartz space, $\mathscr{S}(\mathbb{R})$ - also a Fréchet-Montel space. Or one uses the space of smooth functions with compact support, $C_c^{\infty}(\mathbb{R})$. That is not a metrisable space, but still a Montel space, so it has nice properties despite not being – 2017-02-05
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    metrisable. What the most appropriate space to use is depends on what one wants to do. If you want to integrate over all of $\mathbb{R}$, possibly with weights, then you need fast enough decay at $\infty$, so you'll probably look at $\mathscr{S}$ or $C_c^{\infty}$. If you're more concerned with local behaviour, $C^{\infty}(\mathbb{R})$ is often natural. – 2017-02-05
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    What kind of a metric would I give $C(\Bbb R)$ and $C^\infty(\Bbb R)$? – 2017-02-05
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    Usually, you don't work with a metric on non-normable Fréchet spaces, you use the seminorms. In case of $C(\mathbb{R})$ one often uses $p_k(f) = \max \{ \lvert f(x)\rvert : \lvert x\rvert \leqslant k\}$ (more generally, if $\Omega$ is an open subset of a manifold, one takes an exhaustion $K_n$ of $\Omega$ by compact sets, so that $K_n \subset \operatorname{int} K_{n+1}$ and $\Omega = \bigcup K_n$ and lets $p_n(f)= \max \{ \lvert f(x)\rvert : x \in K_n\}$). – 2017-02-05
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    For $C^{\infty}$, one uses the same seminorms also for the derivatives. Often used are the families $q_{k,n}(f) = p_k\bigl(f^{(n)}\bigr)$ or $\tilde{q}_k(f) = \max \{ p_k(f^{(n)}) : n \leqslant k\}$. If for some reason one needs a metric and the knowledge that one exists isn't sufficient, one usually uses something like $$d(f,g) = \sum_{k = 1}^{\infty} 2^{-k} \frac{p_k(f-g)}{1 + p_k(f-g)}$$ where $(p_k)$ is a sequence of seminorms generating the topology. – 2017-02-05