Definition:Test Function Space

Definition

Denote with $\map \DD {\R^d}$ the set:

$\set { \phi: \R^d \to \C, \text{$\phi$is a test function} }$

of all test functions $\phi: \R^d \to \C$.

Then $\map \DD {\R^d}$ is called the space of test functions.

$\map \DD {\R^d}$ is a vector space over $\C$, i.e.:

$\forall \psi, \phi \in \map \DD {\R^d} : \phi + \psi \in \map \DD {\R^d}$.
$\forall \phi \in \map \DD {\R^d} : \forall \alpha \in \C : \alpha \cdot \phi \in \map \DD {\R^d}$.

$\map \DD {\R^d}$ also has a topology.