Notion of convergence on the space of compactly supported continuous functions 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 
$$ \Vert \varphi \Vert = \sup_{x \in \mathbb{R}^m} \Vert \varphi(x) \Vert.$$
First question: is $E$ equipped with this norm a Banach space?
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 $\varphi \in E$ 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 
$$ \text{supp}(\varphi) \cup \bigcup_{k \in \mathbb{N}} \text{supp}(\varphi_k) \subseteq K,$$
that is, if the supports of the functions do not "escape" to infinity.
Second question: does this notion of convergence somehow generates a topology on $E$? Is it metrizable? 
 A: 
First question: is $E$ equipped with this norm a Banach space?

No.  In fact, its completion is the space of continuous functions $\mathbb{R}^n\to\mathbb{R}^{m}$ that vanish at $\infty$ (that is, $f(x)\to 0$ as $\|x\|\to\infty$).  Given any continuous function $f$ which vanishes at $\infty$, let $f_k$ be obtained by "cutting off" $f$ to be supported on a ball of radius $k$ around the origin (so, say, $f_k$ agrees with $f$ on a ball of radius $k-1$, and interpolates radially to $0$ between the sphere of radius $k-1$ and the sphere of radius $k$).  Then $f_k\in E$ for each $k$ and $f_k$ converges uniformly to $f$, and it follows easily that $(f_k)$ is Cauchy in $E$ but has no limit in $E$ unless $f$ has compact support.

Second question: does this notion of convergence somehow generates a topology on $E$? Is it metrizable? 

There is no metrizable topology which induces this notion of convergence.  To prove this, for each $R$, choose a sequence $(\varphi_k^R)_k$ such that each $\varphi_k^R$ has a ball of radius $R$ as its support but $\varphi_k^R\to 0$ uniformly.  Then each of these sequences $(\varphi_k^R)$ converges to $0$ in Maggi's sense.  If this notion of convergence came from a metric, we could choose for each $R$ a $k_R$ such that $\varphi_{k_R}^R$ is within $1/R$ of $0$ with respect to the metric, and then $\varphi_{k_R}^R$ would converge to $0$ as $R\to\infty$.  But this is impossible since convergent sequences are required to have some uniform compact support.
However, this convergence does come from a natural topology.  For each compact $K\subset\mathbb{R}^n$, let $E_K$ be the subspace of $E$ consisting of functions with support contained in $K$. Say that a set $U\subseteq E$ is a basic neighborhood of $0$ if $U$ is balanced and convex, and $U\cap E_K$ is an open neighborhood of $0$ in $E_K$ with respect to the sup norm on $E_K$ for each $K$.  Finally, say that a set $U\subseteq E$ is open if it is a union of translates of basic neighborhoods of $0$.  
It can be shown that this topology makes $E$ as the colimit of the spaces $E_K$ with their sup norms in the category of locally convex topological vector spaces.  That is, it is the finest topology on $E$ which makes the inclusion maps $E_K\to E$ all continuous and makes $E$ a locally convex topological vector space.  It is then clear that any sequence which converges in Maggi's sense converges in this topology, since such a sequence will converge in $E_K$ with respect to the sup norm.
Proving the converse (every sequence convergent in this topology converges in Maggi's sense) is a little messier, but here's the idea.  Suppose $(f_k)$ is a sequence of functions in $E$ such that the supports of the $f_k$ are not contained in any fixed compact set.  Then, passing to a subsequence of $(f_k)$ if necessary, we can find a sequence $(x_k)$ going to $\infty$ in $\mathbb{R}^n$ such that $f_k(x_k)\neq 0$ for each $k$.  Now let $U$ be the set of $f\in E$ such that $|f(x_k)|<|f_k(x_k)|$ for each $k$.  Then $U$ is a basic neighborhood of $0$ in $E$ (this uses the fact that any $K$ contains only finitely many of the $x_k$).  Since $f_k\not\in U$ for all $k$, this means $(f_k)$ cannot converge to $0$.
