On docile spaces, Mackey first countable spaces, and sequentially Mackey first countable spaces

In this article we discuss the relationship between three types of locally convex spaces: docile spaces, Mackey first countable spaces, and sequentially Mackey first countable spaces. More precisely, we show that docile spaces are sequentially Mackey first countable. We also show the existence of sequentially Mackey first countable spaces that are not Mackey first countable, and we characterize Mackey first countable spaces in terms of normability of certain inductive limits.