Definition 4

The dual space of $\mathscr{S}(\mathbf{R})$ endowed with the topology of bounded convergence is called the space of tempered distributions on $\mathbf{R}^n$. It is denoted by $\mathscr{S}'(\mathbf{R}^n)$.

Since $\mathscr{S}(\mathbf{R}^n)$ is bornological, the space $\mathscr{S}'(\mathbf{R}^n)$ is complete (EVT, III, p. 24, cor. 1). Since $\mathscr{S}(\mathbf{R}^n)$ is a Montel space, the same is true of $\mathscr{S}'(\mathbf{R}^n)$ (EVT, IV, p. 19, prop. 9). The space $\mathscr{S}'(\mathbf{R}^n)$ is therefore reflexive (EVT, IV, p. 16, th. 2).