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Let $\tau_1$ and $\tau_2$ be locally convex Hausdorff topologies on vector space $X$ such that $(X,\tau_1)^\ast = (X,\tau_2)^\ast$. It is well known that $(X,\tau_1)$ and $(X,\tau_2)$ have the same convex closed sets. Does that keep if $\tau_1$ and $\tau_2$ are vector, not necessarily locally convex topologies?

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