Topological properties of strict $(LF)$-spaces and strong duals of Montel strict $(LF)$-spaces

Following [2], a Tychonoff space $X$ is Ascoli if every compact subset of $C_k(X)$ is equicontinuous. By the classical Ascoli theorem every $k$-space is Ascoli. We show that a strict $(LF)$-space $E$ is Ascoli iff $E$ is a Fr'{e}chet space or $E=\varphi$. We prove that the strong dual $E_{\beta }^{\mathrm{\prime }}$ of a Montel strict $(LF)$-space $E$ is an Ascoli space iff one of the following assertions holds: (i) $E$ is a Fr'{e}chet–Montel space, so $E_{\beta }^{\mathrm{\prime }}$ is a sequential non-Fr'{e}chet–Urysohn space, or (ii) $E=\varphi$, so $E’_\beta= $\mathbb{R}$^\omega$\mathrm{.}Consequently,thespace$$\mathcal{D}$(\Omega)$ of test functions and the space of distributions ${\mathcal{D}}^{\mathrm{\prime }}(\mathrm{\Omega })$ are not Ascoli that strengthens results of Shirai [20] and Dudley [5], respectively.