Is there an example of a locally convex topological vector space which is not [compactly generated](https://en.wikipedia.org/wiki/Compactly_generated_space)?


(any such example must be non-Fréchet, since [all Fréchet spaces are compactly generated](https://mathoverflow.net/a/52749/238))


(note: I am using "compactly generated" in the topologists' sense, not the functional analysts' sense)



Example: a locally convex TVS which is not compactly generated
If $A$ is uncountable then Problem J(b), p. 240 of [Kelley](http://books.google.com/books?id=-goleb9Ov3oC&pg=PA240) says that $\mathbf R^A$ (or $\mathbf C^A$) with the product topology (a.k.a. uniform convergence on finite subsets) is not compactly generated. This topology is locally convex, being generated by the seminorms $\|z\|\_F=\sup\_{a\in F}|z\_a|$ for all finite $F\subset A$.


**Edit:** Also, if I'm not mistaken, Frölicher & Roulin's [*Topologies faibles et topologies à génération compacte*](http://retro.seals.ch/digbib/view?rid=ensmat-001:1972:18%3a:354) shows that an infinite-dimensional separable Hilbert space with its weak topology is never compactly generated; and its [Math Review](http://www.ams.org/mathscinet-getitem?mr=333647) adds that this remains true of the dual of any infinite-dimensional Banach space, with the weak\* topology.


