TITLE: Is every Montel locally convex vector space compactly generated?
QUESTION [5 upvotes]: 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?

REPLY [5 votes]: An(other) example of a Montel space which is not compactly generated is $\kern.4mm\mathscr D\kern.4mm(\kern.4mm\mathbb R\kern.4mm)$ . This follows from Theorem 6.1.4(iii) and Proposition 6.2.8(ii) on pages 190 and 195 in 
A. Frölicher and A. Kriegl: Linear Spaces and Differentiation Theory, Wiley, Chichester 1988.