Which topology for $C^\infty(X)$ works?

Question

Let $X$ be a smooth manifold. What is the appropriate topology on $C^\infty(X)$ such that a linear functional $\lambda$ on $C^\infty(X)$ is continuous iff it can be represented as a limit of the form $$\lambda f = \lim_{\ell\to\infty} \sum_{k=1}^{n_\ell}\lambda_{k\ell}f(x_{k\ell})$$ with finite $n_\ell$, suitable numbers $\lambda_{k\ell}$, and suitable points $x_{k\ell}\in X$?

Answer 1

Arnold, if $l$ is not obliged to belong to $\mathbb N$, i.e. the index set for $l$ can be arbitrary (for $l\in\mathbb N$ this seems to be also true, but this requires a verification), then the usual (the weakest locally convex) topology of $C^\infty(M)$ generated by the projections to the spaces $C^\infty(U)$, where $U$ is diffeomorphic to ${\mathbb R}^n$ (or to an open subset in ${\mathbb R}^n$, the topology on such $C^\infty(U)$ is described in the Kirillov-Gvishiani book), satisfies your requirement, since the linear combinations of delta-functionals are dense in $(C^\infty(M))'$ (this in its turn follows from the fact that delta-functionals separate elements of $C^\infty(M)$).

In this topology a net of functions $f_\nu$ tends to a function $f$ in $C^\infty(M)$ if and only if for each subset $U\subseteq M$, diffeomorphic to $\mathbb R^n$, the restrictions $f_\nu\big|_U$ tend to $f\big|_U$ uniformly on each compact set $K\subseteq U$ by each partial derivative $\partial^m$ generated by the local chart for $U$: $$ \forall m\in{\mathbb N}^n\qquad \sup_{x\in K}\bigg|\partial^m\Big(f_\nu\big|_U-f\big|_U\Big)(x)\bigg|\underset{\nu\to\infty}{\longrightarrow}0 $$ (I assume everywhere that $\mathbb N$ contains zero).

There is another "abstract way" to describe this topology that I find "more visual", I told about it here:

1. For each function $f\in {\mathcal C}^\infty(M)$ let us define its support as the closure of the set of the points where $f$ does not vanish: $$ \text{supp}f=\overline{\{x\in M:\ f(x)\ne 0\}}. $$ An equivalent definition: $\text{supp}f$ is the set of the points in $M$ where $f$ has non-zero germs: $$ \text{supp}f=\{x\in M:\ f\not\equiv 0\ (\text{mod}\ x)\}. $$

2. Let us define differential operators (see e.g. S.Helgason's book) on $M$ as linear mappings $D:{\mathcal C}^\infty(M)\to {\mathcal C}^\infty(M)$ which do not extend the support of functions: $$ \text{supp}Df\subseteq \text{supp}f,\quad f\in{\mathcal C}^\infty(M). $$ Equivalently, $D$ is local, i.e. the value of $Df$ in a point $x\in M$ depends only on the germ of $f$ in $x$: $$ \forall f,g\in{\mathcal C}^\infty(M)\quad \forall x\in M\qquad f\equiv g\ (\text{mod}\ x)\quad\Longrightarrow\quad Df(x)=Dg(x). $$

3. Then we say that a sequence of functions $f_n$ converges to a function $f$ in ${\mathcal C}^\infty(M)$ $$ f_n\overset{{\mathcal C}^\infty(M)}{\underset{n\to\infty}{\longrightarrow}}f $$ if and only if for each differential operator $D:{\mathcal C}^\infty(M)\to {\mathcal C}^\infty(M)$ the sequence of functions $Df_n$ converges to $Df$ in the space ${\mathcal C}(M)$ of continuous functions with the usual topology of uniform convergence on compact sets in $M$: $$ Df_n\overset{{\mathcal C}(M)}{\underset{n\to\infty}{\longrightarrow}}Df $$

The definition of the differential operator in this construction belongs (as far as I know) to Jaak Peetre. We can replace it by the one that Jet Nestruev uses for abstract rings, and the topology on $C^\infty(M)$ will be the same. Similarly, we can replace $D$ by arbitrary compositions of vector fields on $M$ $$ D=X_1\circ...\circ X_p, \quad p\in {\mathbb N}, $$ where vector fields are defined as derivations of the ring $C^\infty(M)$, i.e. linear operators $X:C^\infty(M)\to C^\infty(M)$ with the Leibnitz property $$ X(f\cdot g)=X(f)\cdot g+f\cdot X(g). $$