Ultraregular inductive limits (Q749818)

Let \(E_ 1\subset E_ 2\subset..\). be a sequence of locally convex spaces with continuous inclusions and E the corresponding inductive limit. E is said to be ultraregular if each bounded subset of E is contained and bounded in some \(E_ n\). It is proved that if each \(E_ n\) is closed in \(E_{n+1}\) and each \(E_ n\) is an LF-space, then E is ultraregular if and only if every absolutely convex closed neighborhood of \(E_ n\) is closed in \(E_{n+1}\).