Exercise 4, § 2

Let $E$ be a Hausdorff prehilbertian space, $(E_i)_{i \in I}$ a family of complete vector subspaces of $E$, well-ordered by inclusion, such that the union of all $E_i$ is everywhere dense in $E$. Show that there exists an orthonormal basis $(e_\alpha)_{\alpha \in A}$ in $E$ with the following property : for every $i \in I$, the set of all $e_\alpha$ belonging to $E_i$ is an orthonormal basis of $E_i$. (Consider the set of all orthonormal subsets $S$ in $E$ such that, for every $i \in I$, every vector of $S$ not belonging to $E_i$, is orthogonal to $E_i$, and take a maximal element of this set.) From this, deduce a new proof of the corollary of V, p. 24.