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 Fréchet space or $E=\phi$. We prove that the strong dual $E'_\beta$ of a Montel strict $(LF)$-space $E$ is an Ascoli space iff one of the following assertions holds: (i) $E$ is a Fréchet--Montel space, so $E'_\beta$ is a sequential non-Fréchet--Urysohn space, or (ii) $E=\phi$, so $E'_\beta= \mathbb{R}^\omega$. Consequently, the space $\mathcal{D}(\Omega)$ of test functions and the space of distributions $\mathcal{D}'(\Omega)$ are not Ascoli that strengthens results of Shirai [20] and Dudley [5], respectively.