Let \( X \subset \mathbb{R}^d \) be an open set. A linear form \( \nu \colon C_c^\infty(X) \to \mathbb{C} \) is called a distribution if, for every compact set \( K \subset X \), there is a real number \( C \geq 0 \) and a non-negative integer \( N \) such that for all \( \phi \in C_c^\infty(X) \) with \( \text{supp } \phi \subset K \),

\begin{equation} \lvert \langle \nu, \phi \rangle \rvert \leq C \displaystyle{\sum_{\lvert \alpha \rvert \leq N} \sup \lvert \partial^\alpha \phi \rvert} \end{equation}

The vector space of distributions on \( X \) is denoted by \( \mathcal{D}’(X) \).

The support of a distribution \( \nu \in \mathcal{D}’(X) \), written \( \text{supp }\nu \), is by definition the complement of the set \( \big\{ x \in X : \nu = 0 \text{ in a neighborhood of } x \big\} \).

Sequencial continuity.

A linear form \( \nu \) on \( C_c^\infty(X) \) is a distribution if and only if \( \lim_n \langle \nu, \phi_n \rangle = 0 \) for every sequence \( \big\{ \phi_n \big\}_{n \in \mathbb{N}} \) which converges to zero in \( C_c^\infty(X) \).

Localization

Let \( X \subset \mathbb{R}^d \) be an open set, and let \( \big\{ X_\lambda \big\}_{\lambda \in \Lambda} \) be an open cover of \( X \) (where \( \Lambda \) is an index set).  Suppose that, for each \( \lambda \in \Lambda \) there is a distribution \( \nu_\lambda \in \mathcal{D}’(X_\lambda) \) and that

\begin{equation} \nu_\lambda = \nu_\mu \text{ on } X_\lambda \cap X_\mu \text{ if } X_\lambda \cap X_\mu \neq \emptyset. \end{equation}

Then there is a unique distribution \( \nu \in \mathcal{D}’(X) \) such that \( \nu = \nu_\lambda \) in \( X_\lambda \) for each \( \lambda \in \Lambda. \)

Influence of supports

Let \( \nu \in \mathcal{D}’(X) \) and let \( \phi \in C_c^\infty(X). \) If the supports of \( \nu \) and \( \phi \) are disjoint, then \( \langle \nu, \phi \rangle = 0. \)