Let $X$ be a separable Frechet space (= Polish locally convex linear metric space) and $X'\_c$ be the space of linear continuous functionals on $X$, endowed with the compact-open topology (= the topology of uniform convergence on compact subsets of $X$). 



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> **Question.** Is the space $X\_c'$ sequential (equivalently, a $k$-space)? 
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Is the compact-open topology on the dual of a separable Frechet space sequential?
Yes. In the next paragraph I will show that if $X$ is a Fréchet space (without requiring separability) then $X'\_c$ with the compact-open topology is a $k$-space. As you note, this implies sequentiality if $X$ is separable. 


The compact-open topology on $X'\_c$ is the same as the finest topology agreeing with $\sigma(X'\_c, X)$ on equicontinuous sets by the "Banach-Dieudonné" theorem (see Schaefer's *Topological Vector Spaces*, Chapter IV, Theorem 6.3). As Fréchet spaces are barrelled, all $\sigma(X'\_c,X)$-bounded sets, and therefore all $\sigma(X'\_c,X)$-compact sets, are equicontinuous. We also have, by the Bourbaki-Alaoglu theorem, that closed equicontinuous sets are $\sigma(X'\_c,X)$-compact. Therefore the compact-open topology is the finest topology agreeing with $\sigma(X'\_c,X)$ on compact sets, *i.e.* the Kelleyfication of the $\sigma(X'\_c,X)$ topology, so $X'\_c$ is a $k$-space.


