Locally Integrable Function defines Distribution

Theorem
Let $f \in \map {L^1_{loc}} {\R^d}$ be a locally integrable function.

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

Let $T_f : \map \DD {\R^d} \to \C$ be a mapping.

Then $T_f$ is a distribution.

Existence
Let $\phi \in \map \DD {\R^d}$ be a test function.

Let $T_f$ be defined as


 * $\ds T_f = \int_{\R^d} \map f {\mathbf x} \map \phi {\mathbf x} \rd \mathbf x$

By definition, $\phi$ has the compact support.

Together with the properties of $f$ we have that $T_f$ is bounded with respect to any compact range of integration.

Linearity
By Linear Combination of Integrals we have that $T_f$ is a linear mapping.

Continuity
By Convergent Sequence Minus Limit, we can shift the sequence to set its limit to zero.

Let $\mathbf 0 : \R^d \to 0$ be the zero mapping.

Let $\phi_n$ converge to $\mathbf 0$ in $\map \DD {\R^d}$:


 * $\phi_n \stackrel \DD {\longrightarrow} {\mathbf 0}$

Let $K \subseteq \R^d$ be the compact support of all $\phi_n$.

By definition, $T_f$ is a distribution.