Exercise 2, § 2

a) Let $E_1, E_2$ be two hilbertian spaces whose respective hilbertian dimensions are two cardinals $m$ and $n$ such that $m < n \leq m^{N_0}$. Let $E = E_1 \oplus E_2$ be the hilbertian sum of $E_1, E_2$ and let $(b_\lambda)_{\lambda \in L}$ be an orthonormal basis of $E_2$. Show that there exists an algebraically independent system $(a_\lambda)_{\lambda \in L}$ in $E_1$ (cf. exerc. 1, c)); let $H$ be the subspace of $E$ generated (algebraically) by the family $(a_\lambda + b_\lambda)_{\lambda \in L}$. Show that the hilbertian dimension of $\overline{H}$ is equal to $H$ (observe that the orthogonal projection from $H$ onto $E_2$ is everywhere dense, and use exerc. 1, a)). If $S$ is an orthonormal subset of $H$, show that $S \cap E_2 = \varnothing$; and deduce that $\mathrm{Card}(S) \leq m$ (observe that every element of an orthonormal basis of $E_1$ is orthogonal to every element of $S$ except at most to a countably infinite subset). b) Let $E_3$ be a hilbertian space with hilbertian dimension $p \geq n$, and let $F$ be the hilbertian sum $E \oplus E_3$. Let $G$ be the subspace $H + E_3$ of $F$. Show that the hilbertian dimension of $\overline{G}$ is $p$. If $T$ is an orthonormal subset of $G$, show that $T \cap (E_2 + E_3) \subset E_3$; deduce that the cardinality of the orthogonal projection from $T$ onto $E_2$ is at most $m$ (argue as in a)). Conclude from this that $G$ does not have an orthonormal basis, by observing that the orthogonal projection from $G$ onto $E_2$ is everywhere dense in $E_2$.