Exercise 20, § 2

(a) Now prove that a discretely linearly compact module cannot be a direct sum of an infinity of submodules not reduced to 0.

(b) Give an example of a minimal linear topology (exerc. 17) which is not linearly compact (consider an infinite direct sum of pairwise non-isomorphic simple modules).

(c) Let $E$ be a linearly compact module such that there exists a family of simple submodules whose sum is dense in $E$. Now prove that: 1° $E$ is strictly linearly compact (apply a) to the quotient of $E$ by an open submodule) ; 2° $E$ is isomorphic to the (topological) product of a family of discrete simple modules. (Consider the sets $\mathfrak{D}$ of maximal open submodules of $E$ such that for every finite sequence $(G_k)_{1 \leq k \leq n}$ of $n$ distinct elements of $\mathfrak{D}$, $E / (\bigcap_{k=1}^n G_k)$ is a direct sum of $n$ simple submodules. Now prove that there exists a maximal set $\mathfrak{D}$ (for the relation of inclusion), and that the intersection of all the submodules $G$ belonging to such a set is reduced to 0 ; one will use the fact that $\{0\}$ is intersection of maximal open submodules, and that for every maximal open submodule $G$ of $E$, there is at least one simple submodule of $E$ not contained in $G$. Conclude using the fact that $E$ is strictly linearly compact.)

(d) Deduce from c) that every linearly compact vector space over a field $K$ is strictly linearly compact and isomorphic to a product $K_s^1$.