Let $X$ be a Hausdorff locally convex vector space. Recall (my reference is the book of H. Jarchow, Locally Convex Spaces. B.G. Teubner, 1981) that we say that $X$ is a semi-Montel space if every bounded subset of $X$ is relatively compact (equivalently, every closed and bounded subset of $X$ is compact), and a Montel space if it is semi-Montel and satisfy one (hence all) of the following conditions (equivalent under the semi-Montel hypothesis, see Proposition 11.5.1, pp. 230 of Jarchow's book):
- $X$ is reflexive;
- $X$ is barrelled;
- $X$ is quasi-barrelled.
It is known that the strong dual of a Montel space is also Montel (Jarchow, Proposition 11.5.4, pp. 230-231). In the proof of Theorem 4.11 (5), pp. 39-40 of the book of A. Kriegl and P.W. Michor, The Convenient Setting of Global Analysis (AMS, 1997), it is shown that if $X$ is the strong dual of a Fréchet-Montel space (hence $X$ is a Montel space), then $X$ is a compactly generated topological space (also called a k-space or a Kelley space), i.e. the topology of $X$ is the final topology with respect to the inclusions of compact subsets of $X$. However, it seems to me that the proof of this assertion uses only the fact that $X$ is Montel.
Question: Are Montel spaces compactly generated, or is there a counter-example to this claim?