Category:Definitions/Examples of Tempered Distributions

This category contains definitions of examples of Tempered Distribution.

Let $\map \SS \R$ be a Schwartz space.

Let $\phi, \psi \in \map \SS \R$ be Schwartz test functions.

Let $\alpha \in \C$ be a complex number.

Let $\sequence {\phi_n}_{n \mathop \in \N} \in \map \SS \R$ be a convergent sequence with the limit $\mathbf 0 \in \map \SS \R$.

Suppose a mapping $T : \map \SS \R \to \C$ is linear and continuous:

$\forall \psi, \phi \in \map \SS \R : \map T {\phi + \psi} = \map T \phi + \map T \psi$

$\forall \phi \in \map \SS \R : \forall \alpha \in \C : \map T {\alpha \cdot \phi} = \alpha \cdot \map T \phi$

$\paren {\phi_n \stackrel \SS {\longrightarrow} \mathbf 0} \implies \paren {\map T {\phi_n} \to \map T {\mathbf 0}}$

Then $T$ is a tempered distribution.