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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 isAscoli. We show that a strict $(LF)$-space $E$ is Ascoli iff $E$ is aFr\'{e}chet space or $E=\phi$. We prove that the strong dual $E'_\beta$ of aMontel strict $(LF)$-space $E$ is an Ascoli space iff one of the followingassertions holds: (i) $E$ is a Fr\'{e}chet--Montel space, so $E'_\beta$ is asequential non-Fr\'{e}chet--Urysohn space, or (ii) $E=\phi$, so $E'_\beta=\mathbb{R}^\omega$. Consequently, the space $\mathcal{D}(\Omega)$ of testfunctions and the space of distributions $\mathcal{D}'(\Omega)$ are not Ascolithat strengthens results of Shirai [20] and Dudley [5], respectively.

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