Proposition 3

— Let $X$ be a locally compact space, $X'$ the compact space obtained by adjoining to $X$ a point at infinity $\omega$ (GT, I, §9, No. 8,

Th. 4), and E a Banach space. The closure of $\mathscr{K}(X;E)$ in the normed space $C^b(X;E)$ is the vector space of continuous functions on X, with values in E and tending to 0 at the point $\omega$.

Let $f \in C^b(X;E)$ be a function in the closure of $\mathscr{K}(X;E)$; for every $\varepsilon > 0$, there exists a function $g \in \mathscr{K}(X;E)$ such that $\|f(x) - g(x)\| \leq \varepsilon$ for all $x \in X$; if K is the support of g, it follows that $\|f(x)\| \leq \varepsilon$ for all $x \in \mathbf{C}K$, thus $f(x)$ tends to 0 as $x$ tends to $\omega$. Conversely, if f has this property then, for every $\varepsilon > 0$, there exists a compact set $K \subset X$ such that $\|f(x)\| \leq \varepsilon$ for all $x \in \mathbf{C}K$. By Lemma 1 there exists a continuous mapping h of X into $[0,1]$, with compact support, equal to 1 on K; then $\|f(x)h(x)\| \leq \varepsilon$ on $\mathbf{C}K$ and $f(x) = f(x)h(x)$ on K; since $fh$ has compact support and $\|f(x) - f(x)h(x)\| \leq 2\varepsilon$ for all $x \in X$, the proposition is proved.

We shall denote by $C^0(X;E)$ the subspace of $C^b(X;E)$ formed by the functions tending to zero at the point at infinity $\omega$; it is thus the completion of the normed space $\mathscr{K}(X;E)$.