Let $X$ be a locally convex space. Then as far as I understand the bidual of $X$ with the weak topology, $(X_\beta')_\sigma'$, is like a quasi-completion of $X_\sigma$. Namely, if $B \subseteq X$ is weakly bounded, then $\bar{B} \subseteq (X_\beta')_\sigma'$ is complete. My question is then if $(X_\beta')_\sigma'$ itself is quasi-complete?
2026-03-28 03:52:51.1774669971
Weak quasi-completion of a locally convex space
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Thanks to @Jochen for pointing me in the direction where to look I was able to find a reference for a counterexample. In "Sur les Espaces (F) et (DF)", Summa Brasil. Math. 3 (1954), at p. 88 Grothendieck gives an example of a non-distinguished space $E$ where $E''$ is not weakly quasi-complete.