# Reference request: $\mathcal{C}^\infty_c(M)$ is a topological vector space with the Whitney topology

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.

• 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. Jul 8, 2015 at 15:44

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).