# Continuity and sequential continuity of a linear functional

Let $E = C_c^0(\mathbb{R}^n;\mathbb{R}^m)$ be the space of compactly supported continuous functions on $\mathbb{R}^n$ with values on $\mathbb{R}^m$. There is a natural norm on this space: given $\varphi \in E$, we put $$\Vert \varphi \Vert = \sup_{x \in \mathbb{R}^m} \Vert \varphi(x) \Vert.$$

For each compact set $K \subset \mathbb{R}^n$, let $E_K$ be the subspace of $E$ consisting of those functions that have support contained in $K$. Each $E_K$ inherits a the supremum norm from $E$.

Define a topology $\tau$ on $E$ by declaring a subset $U$ to be open if $U \cap E_K$ is open in $E_K$ for each compact set $K \subset \mathbb{R}^n$.

In the book "Sets of finite perimeter and geometric variational problems", by Francesco Maggi, the author introduces the following notion of convergence on $E$: a sequence $(\varphi_k)_{k \in \mathbb{N}}$ in $E$ converges to a function $\varphi \in E$ with uniform supports, and we write $\varphi_k \overset{us}{\to} \varphi$, if $\Vert \varphi_k - \varphi \Vert \to 0$ as $k \to \infty$ and if there is a compact set $K \subset \mathbb{R}^n$ such that

$$supp(\varphi) \cup \bigcup_{k \in \mathbb{N}} supp(\varphi_k) \subseteq K,$$

that is, if the supports of the functions do not "escape" to infinity.

Now let $L : E \to \mathbb{R}$ be a linear functional.

Question:

Is it true that $L : (E, \tau) \to \mathbb{R}$ is continuous if and only if $L(\varphi_k) \overset{us}{\to} L(\varphi)$ whenever $\varphi_k \overset{us}{\to}{\varphi}$ in $E$?