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

Abstract

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 Frechet 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 Frechet–Montel space, so $$E'_\beta $$ is a sequential non-Frechet–Urysohn space, or (ii) $$E=\phi $$ . Consequently, the space $$\mathcal {D}(\varOmega )$$ of test functions and the space of distributions $$\mathcal {D}'(\varOmega )$$ 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.