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.
• Suppose the above is true, then it is stated that $F$ is the inductive limit of the $E_j$ because the latter are dense in $F$. I believe this is something I could prove or look up, but I lack a reference for this.

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.

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 subsets 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.
• Suppose the above is true, then it is stated that $F$ is the inductive limit of the $E_j$ because the latter are dense in $F$. I believe this is something I could prove or look up, but I lack a reference for this.

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.
• Suppose the above is true, then it is stated that $F$ is the inductive limit of the $E_j$ because the latter are dense in $F$. I believe this is something I could prove or look up, but I lack a reference for this.