4.19 Support

Definition 4.19.1 (Support). Let $X$ be a topological space, $E$ be a TVS over $K \in \RC$, and $f \in C(X; E)$, then $\supp{f}= \ol{\bracs{f \ne 0}}$ is the support of $f$.

Definition 4.19.2 (Compactly Supported). Let $X$ be a topological space, $E$ be a TVS over $K \in \RC$, and $f \in C(X; E)$, then $f$ is compactly supported if $\supp{f}$ is compact. The set $C_{c}(X; E)$ is the vector space of all $E$-valued compactly supported functions on $X$.

Definition 4.19.3. Let $X$ be a topological space, $f \in C_{c}(X; [0, 1])$ and $U \subset X$ be open, then $f \prec U$ if $\supp{f}\subset U$