Local completion of bornological space

Question

I recently stumbled across this old publication which defines the local completion of a Hausdorff locally convex space. The construction is as follows:

• A Hausdorff locally convex space $E$ is locally complete if every Mackey-Cauchy sequence converges
• Define the local completion $\widetilde{E}$ of $E$ as the intersection of all locally complete subspaces of the ordinary completion $\bar{E}$ that contain $E$
• $\widetilde{E}$ is obviously locally complete

Then the paper claims that if $E$ is bornological, so is $\widetilde{E}$. The proof of this appears rather straightforward but there are two points that seem to elude me:

• Suppose that $(E_j)_{j \in J}$ is a totally ordered (by inclusion) family of bornological subspaces of $\widetilde{E}$ containing $E$. Then it is stated that $F = \cup_{j \in J} E_j$ (I suppose with the subspace topology inherited from $\widetilde{E}$ resp. $\bar{E}$) is a Mackey space. Why would that be true? Note that $E$ is Mackey since it is bornological.

Answer 1

It turns out, that the above is actually rather simple. Clearly, $E$ is dense in $F$ with its subspace topology from $\widetilde{E}$. Furthermore, since $E$ is Mackey, it is well-known that its completion is Mackey as well (e.g. Schaefer, Wolf Topological Vector Spaces, IV.3.5) and the same proof applies here as well. Another more direct way to show this, is to note that $E \to F$ is actually Mackey continuous and by Mackey-Ahrens the Mackey topology on $F$ also induces the original topology on $E$. Hence, the embedding $E \to \bar{E}$ factorises over both $F$ and $(F,\tau(F,F'))$ as subspaces of $\bar{E}$ such that their topologies have to be the same.