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

Following Banakh and Gabriyelyan (Monatshefte Math 180:39–64, 2016), 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= ϕ. We prove that the strong dual Eβ′ 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 Eβ′ is a sequential non-Fréchet–Urysohn space, or (ii) E= ϕ. Consequently, the space D(Ω) of test functions and the space of distributions D ^′ (Ω) are not Ascoli that strengthens results of Shirai (Proc Jpn Acad 35:31–36, 1959) and Dudley (Proc Am Math Soc 27:531–534, 1971), respectively.