Exercise 19, § 2

a) Let $E$ be an $A$-module which is linearly compact. Now prove that the following conditions are equivalent:

α) For every continuous linear mapping $u$ of $E$ into a separated linearly topologized $A$-module $F$, $u$ is a strict morphism (Top. gén., chap. III, 2e éd., § 2, no 8) of $E$ into $F$.

β) For every closed submodule $F$ of $E$, the quotient topology on $E/F$ is a minimal topology (Exercise 17).

γ) $E$ is complete and there is a fundamental system $(U_\lambda)$ of open neighbourhoods of 0 in $E$, formed by submodules, such that the $E/U_\lambda$ are artinian modules.

(To see that γ) implies β), reduce to the case where $F = 0$, and use Exercises 17 and 18.)

When $E$ satisfies the equivalent conditions α), β), and γ), one says that $E$ is strictly linearly compact.

b) Let $E$ be a separated linearly topologized $A$-module, $F$ a closed submodule of $E$. For $E$ to be strictly linearly compact it is necessary and sufficient that $F$ and $E/F$ be so.

c) Any projective limit of strictly linearly compact modules is strictly linearly compact.

d) Let $E$ be a separated linearly topologized $A$-module, $u$ a linear mapping of $E$ into a strictly linearly compact $A$-module $F$. Now prove that if the graph of $u$ is closed in $E \times F$, $u$ is continuous (use Exercise 16 c), and the fact that if $M$ is closed in $E$ and $E/M$ artinian, $M$ is open in $E$.