  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.
