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 $$\langle \nu, \phi \rangle = 0.$$