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 Ck(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 Frechet space or E=phi. We prove that the strong dual E of a Montel strict (LF)-space E is an Ascoli space iff one of the following assertions holds: (i) E is a Frechet-Montel space, so E is a sequential non-Frechet-Urysohn space, or (ii) E=phi. Consequently, the space D() of test functions and the space of distributions D() 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.