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 $E$?
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> 
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Continuity and sequential continuity of a linear functional
The space $E:=C\_c^0(\mathbb{R^n})$ is the inductive limit in the TVS category of the spaces $(E\_K,\|\cdot\|\_{K,\infty})$. As a consequence, a sequence converges on $E$ if and only if it is included in a space $E\_K$ and converges there; a linear form on $E$ is continuous if and only if it is continuous on every $E\_K$, and the claim follows. Note that $E$ is not metrizable, nor first countable, but being an inductive limit of metric TVS, for the above reasons sequential continuity of a linear form is still equivalent to continuity.


