Lemma 2

*Let $X$ be a locally compact space, $K$ a compact subset of $X$, $E$ a locally convex space, $q$ a continuous semi-norm on $E$, and $\Phi$ an equicontinuous set of mappings of $X$ into $E$ whose supports are contained in $K$. Then, for every $\varepsilon > 0$, there exists a continuous partition of unity $(\varphi_j)_{0 \leq j \leq n}$ on $X$ having the following properties: (i) $\operatorname{Supp}(\varphi_j) \subset K$ for $1 \leq j \leq n$. (ii) If, for $1 \leq j \leq n$, $x_j$ is any point of $\operatorname{Supp}(\varphi_j)$, then, for every function $f \in \Phi$ and every $x \in X$,

For every $y$ belonging to the boundary of $K$, one has $f(y) = 0$ for all $f \in \Phi$, therefore there exists an open neighborhood $V_y$ of $y$ in $X$ such that, for every $z \in V_y$ and every $f \in \Phi$, one has $q(f(z)) \leq \varepsilon/2$. Let $K'$ be the set of points of $K$ not belonging to any of the $V_y$ as $y$ runs over the boundary of $K$; $K'$ is compact and is contained in the interior of $K$. The set $\Phi$ is uniformly equicontinuous in $K$; therefore, there exists a finite open covering $(U_j)_{1 \leq j \leq n}$ of $K'$ consisting of open sets in $X$ contained in $K$, such that for every pair of points $x, y$ of a same $U_j$, one has $q(f(x) - f(y)) \leq \varepsilon/2$ for every $f \in \Phi$. By Lemma 1, there exist $n$ continuous mappings $\varphi_j$ of $X$ into $[0, 1]$ ($1 \leq j \leq n$) such that $\operatorname{Supp}(\varphi_j) \subset U_j$ and such that

For $x_j \in \operatorname{Supp}(\varphi_j)$ ($1 \leq j \leq n$) and $f \in \Phi$, we therefore have, for all $x \in U_j$, and this relation remains true if $x \notin U_j$ since then $\varphi_j(x) = 0$. By addition we infer that, for every $x \in X$, where $\varphi_0 = 1 - \sum_{j=1}^n \varphi_j$; whence (1) for $x \in K'$, since then $\varphi_0(x) = 0$; (1) also holds for $x \notin K$, the first member then being zero. Finally, for $x \in K - K'$ we have $q(f(x)\varphi_0(x)) \leq \varepsilon/2$ by the definition of $K'$, therefore this relation and (2) again imply (1) in this case.

Let $X$ be a locally compact space; for every Banach space $E$ (real or complex) we denote by $\mathscr{C}^b(X; E)$ the vector space of continuous and bounded mappings of $X$ into $E$; we know that the topology of uniform convergence in $X$ is compatible with the vector space structure (real, resp. complex) of $\mathscr{C}^b(X; E)$, and it is defined by the norm

Moreover, the normed space thus defined is a Banach space (GT, X, §3, No. 2, and No. 1, Cor. 2 of Prop. 2); the topology defined by this norm on $\mathscr{K}(X; E)$ (in other words, the topology of uniform convergence in $X$) is coarser than the direct limit topology on $\mathscr{K}(X; E)$ defined in No. 1.