Montel (DF)-spaces, sequential (LM)-spaces and the strongest locally convex topology

Abstract

Topologists say that a space is sequential if every sequentially closed set is closed. Directly from the definitions, metrizable ⇒ Fréchet-Urysohn ⇒ sequential ⇒ k-space. Kakol showed that for an (LM)-space (the inductive limit of a sequence of locally convex metrizable spaces), metrizable ⇔ Fréhet-Urysohn. The Cascales and Orihuela result that every (LM)-space is angelic proved that for an (LM)-space, sequential ⇔ k-space. Within the class of (LM)-spaces, then, the four notions become only two distinct ones bearing the relation metrizable ⇒ sequential. Webb proved that every infinite-dimensional Montel (DF)-space is sequential but not Fréchet-Urysohn, and equivalently, not metrizable, since Montel (DF)-spaces are (LB)-spaces and, a fortiori, (LM)-spaces. Does the converse hold in the (LB)-space, (DF)-space or (LM)-space settings? Yes, in all cases. If a (DF)-space or (LM)-space is sequential, then it is either metrizable or it is a Montel (DF)-space. Pfister's result that every (DF)-space is angelic is needed, and the paper provides elementary proofs for this and the similar theorem by Cascales and Orihuela. The strongest locally convex topology plays a vital role throughout.