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?
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Is every Montel locally convex vector space compactly generated?
[Komura's example](http://link.springer.com/article/10.1007%2FBF01361183) mentioned in the comment is just a big product $\mathbb R^{\mathbb R}$ which is a Montel space and its (strong) dual $X$ is thus also Montel. As Komura showed the finest topology $\tau^f$ which agrees on all compact (=equi-continuous) sets with the weak\* (and hence with the strong) topology is not a vector space topology. In
particular, it is different from the strong topology. This should show that $X$ is not
compactly generated.




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The story is quite different if you consider the finest *locall convex* topology such that all inclusions $K\hookrightarrow X$ ($K$ compact) are continuous. This is indeed the associated bornological locally convex topology.


