Proposition 7

Let (M_i) be a family of subsets of E; the submodule orthogonal to the union of the M_i is the intersection of the submodules M'_i which are respectively orthogonal to the M_i; this submodule is also the submodule orthogonal to the submodule of E generated by the union of the M_i.

These results are immediate consequences of the definitions.

There is an analogous proposition (which we shall leave to the reader to state) for submodules of E orthogonal to subsets of E*.

If M is a submodule of E, M' the submodule of E* orthogonal to M and M'' the submodule of E orthogonal to M', then M ⊂ M'' but it may be that M ≠ M'' (Exercise 9). Note however that if M'' is the orthogonal of M'' in E*, then M'' = M'; for M' ⊂ M'' and on the other hand the relation M ⊂ M'' implies M'' ⊂ M'.