Definition 1

— Let X be a topological space, let E be either $\overline{\mathbf{R}}$ or a vector space over $\mathbf{R}$, and let f be a mapping of X into E. The smallest closed set S in X such that $f(x) = 0$ on $X - S$ (in other words, the closure in X of the set of all $x \in X$ such that $f(x) \neq 0$) is called the support of f and is denoted Supp(f).

Let X be a locally compact space, E a topological vector space over $\mathbf{R}$ or $\mathbf{C}$; recall that $\mathscr{C}(X;E)$ denotes the vector space of continuous mappings of X into E; when $E = \mathbf{R}$ or $E = \mathbf{C}$, we will omit the mention of E in this notation if no confusion can result. We shall denote by $\mathscr{K}(X;E)$ the subspace of $\mathscr{C}(X;E)$ formed by the continuous mappings with compact support; for every subset A of X, we denote by $\mathscr{C}(X,A;E)$ (resp. $\mathscr{K}(X,A;E)$) the subspace of $\mathscr{C}(X;E)$ (resp. $\mathscr{K}(X;E)$) formed by the mappings f such that $\operatorname{Supp}(f) \subset A$. If $E = \mathbf{R}$ or $E = \mathbf{C}$, we write $\mathscr{K}(X)$ (resp. $\mathscr{K}(X,A)$) instead of $\mathscr{K}(X;\mathbf{R})$ or $\mathscr{K}(X;\mathbf{C})$ (resp. $\mathscr{K}(X,A;\mathbf{R})$ or $\mathscr{K}(X,A;\mathbf{C})$), provided no confusion can result; we denote by $\mathscr{K}_+(X)$ the pointed convex cone formed by the functions $\geq 0$ of $\mathscr{K}(X;\mathbf{R})$.

For every compact subset K of X, the space $\mathscr{K}(X,K;E)$ may be identified with a subspace of the space of continuous functions $\mathscr{C}(K;E)$ (namely, the subspace of continuous mappings of K into E that are zero on the boundary$^1$ of K). When $\mathscr{C}(K;E)$ is equipped with the topology of uniform convergence in K, $\mathscr{K}(X,K;E)$ is a closed subspace of $\mathscr{C}(K;E)$. In particular, if E is a Fréchet space (resp. a Banach space), then so is $\mathscr{K}(X,K;E)$, because if the topology of E is defined by the semi-norms $p_n$ (resp. the norm $x \mapsto \|x\|$) then the topology of $\mathscr{K}(X,K;E)$ is defined

(1) The original is frontière, also translated as 'frontier' (GT, I, §1, No. 6, Def. 11).

by the semi-norms $f \mapsto \sup_{x \in K} p_n(f(x))$ (resp. the norm $f \mapsto \sup_{x \in K} \|f(x)\|$, denoted $\|f\|$).

The space $\mathscr{K}(X; E)$ is the union of the increasing directed family of subspaces $\mathscr{K}(X, K; E)$, where $K$ runs over the set of compact subsets of $X$; moreover, if $K_1 \subset K_2$ are two compact subsets of $X$, the canonical injection $\mathscr{K}(X, K_1; E) \to \mathscr{K}(X, K_2; E)$ is continuous for the topologies defined above. If $E$ is locally convex, one can therefore define on $\mathscr{K}(X; E)$ the direct limit² of the locally convex topologies of the $\mathscr{K}(X, K; E)$ (TVS, II, §4, No. 4); unless expressly mentioned to the contrary, this will always be the topology in question when we regard $\mathscr{K}(X; E)$ as a topological vector space.