## Distributions. Definition and Basic Properties

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 $\text{supp } \phi \subset K$,

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

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

$\nu_\lambda = \nu_\mu$ on $X_\lambda \cap X_\mu$ if $X_\lambda \cap X_\mu \neq \emptyset$.

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