Exercise 15, § 2

a) Let $E$ be an $A$-module with a linear topology (exerc. 14). In order that a filter basis $\mathscr{B}$ on $E$ formed by affine linear varieties admit an adherent point, it is necessary and sufficient that there exist a convergent filter basis $\mathscr{B}' \supset \mathscr{B}$ formed by affine linear varieties. One says that $E$ is linearly compact if it is separated and if every filter basis on $E$ formed by affine linear varieties admits at least one adherent point. Every artinian module is linearly compact for the discrete topology. Every linearly compact submodule of a separated linearly topologized module $F$ is closed in $F$.

EXERCISES

b) If E is a linearly compact A-module, u a continuous a-linear mapping of E into a separated linearly topologized A-module F, u(E) is a linearly compact submodule of F.

c) Let E be a separated linearly topologized A-module, F a closed submodule of E. In order that E be linearly compact, it is necessary and sufficient that F and E/F be so.

d) Every product of linearly compact modules is linearly compact (consider a maximal filter basis among the filter bases formed by affine linear varieties).

e) Let (E_α, f_{αβ}) be a projective system of linearly topologized modules relative to a filtering set of indices; suppose that the f_{αβ} are continuous a-linear mappings, and that for α ≤ β, f_{αβ}^{-1}(0) is a linearly compact submodule of E_β. Let E = lim E_α, and let f_α be the canonical mapping E → E_α; now prove that for every α, one has

(in particular if the f_{αβ} are surjective, the same is true of the f_α), and f_α^{-1}(0) is linearly compact (use d), and Top.gén., Chapter I, 3rd ed., Appendix, No. 2, Theorem 1).