Remark 10

Let $A$ be a field, $E, F$ two left vector spaces over $A$, $\Phi$ a non-degenerate sesquilinear form on $E \times F$ for an antiautomorphism $J$ of $A$.

a) Let $M$ be a subspace of $E$, $N$ a subspace of $F$ such that $N \supset M^0$ and $M \supset N^0$. Show that if one of the spaces $N/M^0, M/N^0$ is of finite dimension, the same is true of the other, and the dimensions of these two spaces are equal.

b) Let $M, M'$ be two subspaces of $E$ such that $M^{00} = M$ and that $M'$ is of finite dimension; show that one has $(M \cap M')^0 = M^0 + {M'}^0$ and $(M + M')^{00} = M + M'$. (Applying $a$) to the subspaces $M'$ and $M^0 + {M'}^0$, show that $\dim(M \cap M') = \operatorname{codim}(M^0 + {M'}^0)$; applying $a$ to the subspaces $M + M'$ and $M^0$, show that

c) If $E = F$ and if $M$ is a subspace of $E$ such that $E = M^0 + M^{00}$, show that $E$ is direct sum of $M^0$ and $M^{00}$.

d) Let $E$ be a vector space over a commutative field $A$ admitting an infinite countable basis $(e_n)_{n \geqslant 0}$, and let $\Phi$ be the symmetric bilinear form on $E$ such that $\Phi(e_n, e_n) = 1$ for all $n$, $\Phi(e_i, e_j) = 0$ for $i \geqslant 1, j \geqslant 1$ and $i \neq j$, and $\Phi(e_0, e_n) = 1$ for all $n \geqslant 1$. Now prove that $\Phi$ is nondegenerate. Let $M$ (resp. $N$) be the subspace of $E$ generated by the $e_{2k}$ (resp. $e_{2k-1}$) for $k \geqslant 1$, and let $H = M + N$, which is a hyperplane in $E$. Now prove that one has $M^0 = N, N^0 = M, H^{00} = E \neq H, (M \cap N)^0 \neq M^0 + N^0$ and $(M + N)^{00} \neq M + N$, though $M^{00} = M, N^{00} = N$; if $L$ is the subspace of finite dimension 2 generated by $e_0$ and $e_1$, one has $(L \cap H)^0 \neq L^0 + H^0$.