TITLE: Example: a locally convex TVS which is not compactly generated
QUESTION [5 upvotes]: Is there an example of a locally convex topological vector space which is not compactly generated?
(any such example must be non-Fréchet, since all Fréchet spaces are compactly generated)
(note: I am using "compactly generated" in the topologists' sense, not the functional analysts' sense)

REPLY [9 votes]: If $A$ is uncountable then Problem J(b), p. 240 of Kelley 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 shows that an infinite-dimensional separable Hilbert space with its weak topology is never compactly generated; and its Math Review adds that this remains true of the dual of any infinite-dimensional Banach space, with the weak* topology.