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Is there an example of a locally convex Hausdorff space $(X, \tau)$ such that on its dual $X'$ the topology $\tau_{c_0}$ of uniform convergence on $\tau$-null sequences and the topology $\tau_c$ of $\tau$-compact convex sets are not comparable?

Clearly, both topologies are stronger than the weak*-topology and both are weaker than the topology $\tau_k$ of uniform convergence on compact sets.

It is known that

  • if $X$ is metrizable then $\tau_{c_0} = \tau_k$ and
  • if $X$ is complete then $\tau_c = \tau_k$. For this it is enough that $X$ satisfies the convex compactness property meaning that the closed absolutely convex hull of compact sets is compact.

Thus, the example must be necessarily neither metrizable nor satisfying the convex compactness property. So, some weak topology of a certain dual pair should be possibly a candidate for $\tau$, but I can't figure out the dual pair.

Is there also an example for which $\tau_{c_0}$ is compatible but not comparable with $\tau_c$? This is equivalent to saying that $\tau$ satisfies the sequential convex compactness property meaning that the closed absolutely convex hull of null sequences (including the limit $0$) are compact.

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