Category:Definitions/Distributions

This category contains definitions related to Distributions.
Related results can be found in Category:Distributions.

Let $\map \DD {\R^d}$ be a test function space.

Let $\phi, \psi \in \map \DD {\R^d}$ be test functions.

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

Let $\sequence {\phi_n}_{n \mathop \in \N}$ be a convergent sequence in $\map \DD {\R^d}$ with the limit $\phi \in \map \DD {\R^d}$.

Suppose a mapping $T : \map \DD {\R^d} \to \C$ is linear and continuous:

$\forall \psi, \phi \in \map \DD {\R^d} : \map T {\phi + \psi} = \map T \phi + \map T \psi$

$\forall \phi \in \map \DD {\R^d} : \forall \alpha \in \C : \map T {\alpha \cdot \phi} = \alpha \cdot \map T \phi$

$\paren {\phi_n \stackrel \DD {\longrightarrow} \phi} \implies \paren {\map T {\phi_n} \to \map T \phi}$

Then $T$ is a distribution.