Proposition 7

a) The mapping $W\mapsto \mathfrak{a}(W)$ is a bijection from the set of linear subspaces of V to the set of left ideals of A.

b) The mapping $W\mapsto \mathfrak{b}(W)$ is a bijection from the set of linear subspaces of V to the set of right ideals of A.

c) Let $W_1$ and $W_2$ be linear subspaces of V. The relations $W_1\subset W_2$, $\mathfrak{a}(W_1)\supset \mathfrak{a}(W_2)$, and $\mathfrak{b}(W_1)\subset \mathfrak{b}(W_2)$ are equivalent.

Assertion b) follows from Example 1, b) of VIII, p. 104 applied to the invertible $(D^o,A^o$)-bimodule V, as does the equivalence of the relations $W_1\subset W_2$ and $\mathfrak{b}(W_1)\subset \mathfrak{b}(W_2)$.

Let $V^*$ be the dual of V, viewed as a right vector space over the opposite field $D^o$ of D. For any subspace W of V, denote the orthogonal of W in $V^*$ by $W'$. The mapping $W\mapsto W'$ is a bijection from the set of subspaces of V to the set of subspaces of $V^*$. If $W_1$ and $W_2$ are two subspaces of V, then the relations $W_1\subset W_2$ and $W'_1\supset W'_2$ are equivalent. Now, the mapping $u\mapsto^tu$ is an isomorphism from A to the opposite ring of End$_{D^o}(V^*)$; it transforms left ideals of A into right ideals of End$_{D^o}(V^*)$ and $\mathfrak{a}(W)$ into the set $\mathfrak{b}(W')$ of endomorphisms $h$ of $V^*$ such that $h(V^*)\subset W'$. Assertion a), as well as the equivalence of the relations $W_1\subset W_2$ and $\mathfrak{a}(W_1)\supset \mathfrak{a}(W_2)$, then follows from the assertion analogous to b) for the dual $V^*$ of V.