On ultraregular inductive limits (Q5928413)

Let \((E_n, j_n)_{n\in\mathbb{N}}\) be an embedding spectrum of locally convex spaces for which the inductive limit \(\text{ind }_n\) exists. The following assertions are proved:NEWLINENEWLINENEWLINE(a) \(\text{ind }E_n\) has property \((P)\) (i.e., each closed absolutely convex zero-neighborhood in \(E_n\) is closed in \(E_{n+ 1}\)) if and only if each closed convex set in \(E_n\) is closed in \(\text{ind }E_n\).NEWLINENEWLINENEWLINE(b) If \(\text{ind }E_n\) has property \((P)\) then \(\text{ind }E_n\) is ultraregular.NEWLINENEWLINENEWLINE(c) If all the spaces \(E_n\) are Fréchet spaces then the property \((P)\), ultraregularity, and the strictness of the inductive limit are equivalent.