> [!definition] > > A **tempered distribution** is a [[Continuity|continuous]] [[Linear Functional|linear functional]] on the [[Schwartz Space|Schwartz space]] $\cs(\real^d)$. The set $\cs'$ is the space of all tempered distributions, equipped with the [[Weak Topology|weak* topology]]. > > If $F \in \cs'$, then its restriction $F|_{\cd}$ to the [[Space of Test Functions|test functions]] is a [[Distribution|distribution]]. Since $\cd$ is dense in $\cs$, $\cs'$ can be identified as a subspace of $\cd'$ admitting a [[Linear Extension Theorem|continuous extension]] to $\cs$. In particular, the [[Compactly Supported Distribution|compactly supported distributions]] $\ce'$ are tempered. > > If $f \in \loci$, then $f$ is a **tempered function** if the distribution $T_f$ is tempered.