Definition 1

The dual space of $\mathscr{D}(U)$, endowed with the topology of bounded convergence, is called the space of distributions on U. It is denoted by $\mathscr{D}'(U)$.

A distribution on U is therefore a continuous linear form on $\mathscr{D}(U)$.

If $f$ is a linear form on $\mathscr{D}(U)$ (and in particular if $f$ is a distribution) and if $\varphi$ belongs to $\mathscr{D}$(U), we shall denote by $\langle f, \varphi \rangle$ the value of $f$ at $\varphi$.

Since $\mathscr{D}(U)$ is bornological, the space $\mathscr{D}'(U)$ is complete (EVT, III, p. 24, cor. 1). Since $\mathscr{D}(U)$ is a Montel space, the same is true of $\mathscr{D}'(U)$ (EVT, IV, p. 19, prop. 9). In particular, the space $\mathscr{D}'(U)$ is reflexive (EVT, IV, p. 16, th. 2).