A locally convex topological vector space is said to be quasi-complete if its closed bounded subsets are complete. Given a locally convex space $E$, one can show the existence of a Hausdorff quasi-complete locally convex space $\overparen E$ and of a continuous linear map $\overparen\varphi:E\to\overparen E$ statisfying the following universal property :
for every continuous linear map $u$ from $E$ into a Hausdorff quasi-complete locally convex space $F$, there is one and only one continuous linear map $\overparen u:\overparen E\to F$ such that $u=\overparen u\circ\overparen\varphi$.
For example, if $\widehat E$ is a Hausdorff completion of $E$ and $\widehat\varphi:E\to\widehat E$ is the associated map, the space $\overparen E$ can be taken to be the intersection of all quasi-complete linear subspaces of $\widehat E$ containing $\widehat\varphi(E)$ and $\overparen\varphi$ can be taken to be the map induced by $\widehat\varphi$.
Now, I would like to know if this « Hausdorff quasi-completion » can be realized in a little more concrete way. More precisely, there is a theorem of Grothendieck that implies that $\widehat E$ and $\widehat\varphi$ can be realized as follows :
- take for $\widehat E$ the space of all linear functionals on the weak dual $E'_s$ of $E$ whose restriction to any equicontinuous subset of $E'_s$ is continuous and endow it with the topology of uniform convergence on the equicontinuous subsets of $E'_s$,
- take for $\widehat\varphi$ the map $x\mapsto(x'\mapsto x'(x))$.
Within the context of this particular realization of the Hausdorff completion of $E$, is there a nice description of what the intersection of all quasi-complete linear subspaces of $\widehat E$ containing $\widehat\varphi(E)$ looks like?
