Definition:Distribution

Definition

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.

The mapping $\map T \phi$ corresponding to the distribution $T$ is presented by some sources as $\innerprod T \phi$.

To avoid confusion between a distribution and the generating function involved, one can write the function as a subscript.

Suppose we have a distribution $T_f : \map \DD \R \to \C$ and a test function $\phi \in \map \DD \R$.

This would correspond to the following mapping:

$\ds \phi \stackrel {T_f} {\longrightarrow} \int_{-\infty}^\infty \map f x \map \phi x \rd x$

2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): generalized function
2017: Amol Sasane: A Friendly Approach to Functional Analysis ... (previous) ... (next): Chapter $\S 6.1$: A glimpse of distribution theory. Test functions, distributions, and examples