Exercise 19, § 2

(a) Let E be a linearly compact A-module. Show that the following conditions are equivalent:

(α) For every continuous linear mapping $u$ from E to a Hausdorff linearly topologized A-module F, $u$ is a strict morphism (General Topology, Chapter III, § 2, no. 8) from E to 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, consisting of submodules and such that the $E/U_\lambda$ are Artinian modules.

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

If E satisfies the equivalent conditions (α), (β) and (γ), E is called strictly linearly compact.

(b) Let E be a Hausdorff linearly topologized A-module and 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) Every inverse limit of strictly linearly compact modules is strictly linearly compact.

(d) Let E be a Hausdorff linearly topologized A-module and $u$ a linear mapping from E to a strictly linearly compact A-module F. Show 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 $E/M$ is Artinian, M is open in E).