Proposition 15

— Let X be a locally compact space and, for every $x \in X$, let $\varepsilon_x$ be the Dirac measure at the point x. The mapping $x \mapsto \varepsilon_x$ of X into $\mathscr{C}'(X; \mathbf{C})$ is continuous for the topology of compact convergence on $\mathscr{C}'(X; \mathbf{C})$.

Consider a neighborhood of $\varepsilon_{x_0}$ in $\mathscr{C}'(X; \mathbf{C})$ for this topology, which we can suppose to be defined by taking a number $\delta > 0$, a compact subset H of $\mathscr{C}(X; \mathbf{C})$, and considering the set of measures $\mu$ on X with compact support such that $|\mu(f) - \varepsilon_{x_0}(f)| \leq \delta$ for every function $f \in H$. Since H is equicontinuous, there exists a neighborhood U of $x_0$ in X such that the relation $f \in H$ implies $|f(x) - f(x_0)| \leq \delta$ for all $x \in U$, which may also be written $|\varepsilon_x(f) - \varepsilon_{x_0}(f)| \leq \delta$ and proves the proposition.(*)