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

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Following [2], 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 Fréchet space or E=ϕ. We prove that the strong dual E'β 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'β is a sequential non-Fréchet--Urysohn space, or (ii) E=ϕ, so E'β= Rω. Consequently, the space D(Ω) of test functions and the space of distributions D'(Ω) are not Ascoli that strengthens results of Shirai [20] and Dudley [5], respectively.