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Let $M$ be a smooth manifold and let $\mathcal{C}^\infty(M)$ be the space of all smooth real valued functions with the (strong) Whitney topology. This space is a topological vector space iff $M$ is compact.

The maximal topological vector space in $\mathcal{C}^\infty(M)$ for general $M$ is the subspace $\mathcal{C}_c^\infty(M)$ of all compactly supported functions.

I'm looking for a reference that $\mathcal{C}^\infty_c(M)$ is a topological vector space. Continuity of the addition for $\mathcal{C}^\infty(M)$ is e.g. content of Corollary 3.8 of "Stable Mappings and Their Singularities" of Golubitsky and Guillemin. I'm lacking a reference for the fact that the scalar multiplication $$\mathbb{R}\times\mathcal{C}^\infty_c(M)\rightarrow\mathcal{C}^\infty_c(M)$$ is continuous.

Preferably I would like to have a reference without much overload using the construction of the Whitney topology either over local representations in charts or as the induced topology from the one of the jet bundle.

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    $\begingroup$ Not sure about a reference, but the proof is straight forward. Given $u(x)$ with compact support and $m(x) > 0$, we have $|u-(1+\varepsilon)(u+v)| < m$, provided $|v| < m/2$ and $|\varepsilon| < \min(1,\max(|u|/m)^{-1})$, which shows that multiplication is continuous at $(1,u)$. Rescale to get continuity at any $(k,u)$ and treat the case $k=0$ specially in a similar way. $\endgroup$ – Igor Khavkine Jul 8 '15 at 15:44
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This is contained in Michors book "Manifolds of differentiable mappings" (available online here 1). It is a bit hidden (see p. 35 under point 4., the relevant result for scalar multiplication is 4.4.3 earlier), since Michor wants to discuss suitable topologies on all smooth maps not only on the compactly supported functions. Michor constructs the Whitney topology via jet bundles.

A description of the topology in local charts can be found in 2, the topological vector space property is established in Proposition 4.3. By the way: a detailed proof establishing the folklore fact that the local charts approach and the jet bundle approach yield equivalent topologies can be found in appendix C of loc.cit..

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This can done by showing that the restriction of the Whitney topology to $\mathcal{C}^\infty_c(M)$ coincides with the finest linear topology that makes the inclusions $\mathcal{C}^\infty_c(K)\hookrightarrow\mathcal{C}^\infty_c(M)$ continuous for all $\varnothing\neq K\subset M$ compact, where $$\mathcal{C}^\infty_c(K)=\{f\in\mathcal{C}^\infty(M)\ |\ \mathrm{supp}\,f\in K\}$$ is a closed vector subspace of $\mathcal{C}^\infty(M)$. Recall that $\mathcal{C}^\infty(M)$ is a Fréchet space when seen as the projective limit of the sequence of the Banach spaces $\mathcal{C}^m(K_m)$, $m\in\mathbb{N}$, each of them endowed with the initial topology induced by the supremum norm on the Banach space $\Gamma^0(J^m(K_m,\mathbb{R}))$ of continuous sections of $J^m(K_m,\mathbb{R})$ with respect to some fiber Riemannian metric on (say) $J^m(M,\mathbb{R})$ through the $m$-th order jet prolongation map $f\mapsto j^m f$. Here $\{K_m\ |\ m\in\mathbb{N}\}$ is a compact exhaustion of $M$ (i.e. $K_m\subset M$ compact with non-void interior, $K_m\subset\mathring{K}_{m+1}$ for all $m\in\mathbb{N}$, and $\cup^\infty_{m=1}K_m=M$, so that every $K\subset M$ compact is contained in $K_m$ for some $m\in\mathbb{N}$). We use the assumption (as Kriegl and Michor do in their book The Convenient Setting of Global Analysis, AMS, 1997) that $M$ is second countable in order for $M$ to have a compact exhaustion.

An important consequence is that the $m$-th order jet prolongation is a continuous linear map from $\mathcal{C}^\infty(M)$ into $\Gamma^0(J^m(M,\mathbb{R}))$, the latter endowed with the projective limit (Fréchet) topology provided by the restriction maps to each $K_m$, which just happens to coincide with the compact-open topology.

Recalling that point evaluation of functions is continuous in $\mathcal{C}^\infty(M)$, we conclude that for every $\varnothing\neq K\subset M$ compact, $\mathcal{C}^\infty_c(K)$ is a closed subspace of $\mathcal{C}^\infty(M)$. We endow each $\mathcal{C}^\infty_c(K)$ with the induced topology from $\mathcal{C}^\infty(M)$. Once we do this, we clearly have that for every pair $\varnothing\neq K\subsetneq K'\subset M$ of compact subsets the natural inclusion $\mathcal{C}^\infty_c(K)\hookrightarrow\mathcal{C}^\infty_c(K')$ is strict (i.e. $\mathcal{C}^\infty_c(K')\smallsetminus\mathcal{C}^\infty_c(K)\neq\varnothing$) and the topology of $\mathcal{C}^\infty_c(K)$ is the one induced by $\mathcal{C}^\infty_c(K')$ (particularly, such inclusions are continuous and $\mathcal{C}^\infty_c(K)$, being complete, is a closed subspace of $\mathcal{C}^\infty_c(K')$). We can then write $\mathcal{C}^\infty_c(M)$ as the strict countable inductive limit $$\mathcal{C}^\infty_c(M) = \lim_{\overrightarrow{m\in\mathbb{N}}}\mathcal{C}^\infty_c(K_m)\ ,$$ which equals $\cup^\infty_{m=1}\mathcal{C}^\infty_c(K_m)$ as a vector space. A fundamental system of neighborhoods of zero for the inductive limit topology is provided by the sets $$\mathcal{U}_{\underline{U},\underline{k}}=\{f\in\mathcal{C}^\infty_c(M)\ |\ \forall m\in\mathbb{N}\ ,\,j^{k_m}f(K_m\smallsetminus\mathring{K}_{m-1})\subset U_m\}\ ,\quad (K_0=\varnothing)$$ where $\underline{k}=(k_m)_{m\in\mathbb{N}}$ is a strictly increasing sequence of natural numbers and $\underline{U}=(U_m)_{m\in\mathbb{N}}$ is a sequence of sets such that $U_m\subset J^{k_m}(M,\mathbb{R})$ is an open neighborhood of the zero section for each $k\in\mathbb{N}$. It is not difficult to see that the linear topology generated by this fundamental system of neighborhoods of zero is the finest linear topology such that the natural inclusions $i_m:\mathcal{C}^\infty_c(K_m)\hookrightarrow\mathcal{C}^\infty_c(M)$ are continuous. As such, a basis for this topology is obtained by translating the zero neighborhoods $\mathcal{U}_{\underline{U},\underline{k}}$, which just happens to be the intersection of $\mathcal{C}^\infty_c(M)$ with the basis for the Whitney topology of $\mathcal{C}^\infty(M)$ obtained in Lemma 41.11, pp. 437 of the book of Kriegl and Michor, thus establishing our claim.

Remark: in the case that $M=$ a nonvoid open subset of $\mathbb{R}^n$, the above construction of the inductive limit topology of $\mathcal{C}^\infty_c(M)$ is a standard result in most books on locally convex vector spaces and/or distribution theory. See e.g. J. Horváth, Topological Vector Spaces and Distributions (Addison-Wesley, 1966), H. Jarchow, Locally Convex Spaces (B.G. Teubner, 1981) or F. Trèves, Topological Vector Spaces, Distributions and Kernels (Academic Press, 1967).

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