Remark 11

Let $E, E'$ be two left vector spaces over fields $A, A'$ respectively, of dimension $\geqslant 3$; let $\mathfrak{F}(E)$ (resp. $\mathfrak{F}(E')$ be the lattice ordered set (for the relation of inclusion) formed by the finite-dimensional subspaces of $E$ (resp. $E'$).

a) Let $p$ be a mapping from $\mathfrak{F}(E)$ into $\mathfrak{F}(E')$ such that for all $M \in \mathfrak{F}(E)$, $\dim p(M) = \dim M$, and such that for every pair $(M, N)$ of elements of $\mathfrak{F}(E)$, $p(M + N) = p(M) + p(N)$. Now prove that $p$ is injective; if $p$ is bijective, there exists a bijective semilinear mapping $u$ from $E$ into $E'$ such that one has $u(M) = p(M)$ for all $M \in \mathfrak{F}(E)$ (use Exercise 10 of Chapter II, 2nd ed., App. III).

(b) Give an example where $A' = A$ is commutative, $E' = E$ is of finite dimension, and where there exists a mapping $p$ from $\mathfrak{F}(E)$ into itself, such that $\dim p(M) = \dim M, p(M + N) = p(M) + p(N), p(M \cap N) = p(M) \cap p(N)$, but there does not exist any injective semilinear mapping $u$ from $E$ into itself such that $u(M) = p(M)$ for $M \in \mathfrak{F}(E)$. (Consider the case where there exists a superfield $A''$ of $A$ of finite degree and isomorphic to $A$,