Proposition 8

The space $\mathscr{D}(U)$ is dense in $\mathscr{D}'(U)$.

Let $\lambda$ be a linear form on $\mathscr{D}'(U)$ which is zero on $\mathscr{D}(U)$. Since $\mathscr{D}(U)$ is reflexive, there exists a test function $\varphi \in \mathscr{D}(U)$ such that $\lambda (f) =\langle f, \varphi \rangle$ for every $f\in \mathscr{D}'(U)$. Hence

whence $\varphi = 0$. The proposition then follows from the Hahn–Banach theorem (EVT, II, p. 49, cor. 3 (ii)).

For $h\in \mathscr{C}^{\infty}$(U), the transpose of the continuous linear mapping $\varphi \mapsto h\varphi$ of $\mathscr{D}(U)$ into itself is a continuous linear mapping of $\mathscr{D}'(U)$ into itself, which is denoted by $f\mapsto hf$. This definition is justified because if $f$ is the distribution associated with a measure $\nu$ on U, then $hf$ is associated with the measure $h\cdot \nu$. In fact, for every test function $\varphi \in \mathscr{D}$(U), one computes