Exercise 20, § 7

Let E be a vector space.

(a) Show that if M and N are two vector subspaces of E and $M', N'$ the orthogonals of $M$ and $N$ respectively in $E^*$, the orthogonal of $M \cap N$ is $M' + N'$ (cf. § 2, Exercise 9 (a)).

(b) If $E$ is infinite-dimensional, show that there exist hyperplanes $H'$ of $E^*$ such that the subspace of $E$ orthogonal to $H'$ reduces to 0 (consider a hyperplane containing the coordinate forms corresponding to a basis of $E$).

(c) Show that if $E$ is infinite-dimensional there exists an infinite family $(V_i)$ of subspaces of $E$ such that, if $V'_i$ is the subspace of $E^*$ orthogonal to $V_i$, the subspace of $E^*$ orthogonal to $\bigcap_i V_i$ is distinct from $\sum_i V'_i$.

(d) Deduce from (b) that if $E$ is infinite-dimensional, there exists a decomposition $V' \oplus W'$ of $E^*$ as a direct sum, $W'$ being finite-dimensional, for which the sum $V + W$ of subspaces $V, W$ of $E$ orthogonal respectively to $V'$ and $W'$ is distinct from $E$.

(e) For a subspace of $E$ to be finite-dimensional, it is necessary and sufficient that its orthogonal in $E^*$ be of finite codimension.