Regularity of conservative inductive limits (Q1970273)

Let \(E_1\subset E_2 \subset \cdots \) be a sequence of Hausdorff locally convex spaces equipped with topologies \(\tau_n\). The author considers a set of conditions under which the inductive limit \(\text{ind}E_n\) is a regular space. Let us call that \(\text{ind}E_n\) is quasi-regular, if any bounded set in \(\text{ind}E_n\) is bounded in some space \(E_n\), and call conservative if for any linear subspace \(F\) of \(\text{ind}E_n\), it holds \(\text{ind}(F\cap E_n,\tau_n)=(F, \tau)\). The results are Theorem. Any sequentially complete conservative \(\text{ind} E_n\) is quasi regular and if each space \(E_n\) is closed in \(\text{ind}E_n\) then \(\text{ind}E_n\) is regular.