Exercise 1, § 2

Let $B$ be an orthonormal basis in an infinite dimensional hilbertian space $E$. a) Show that every everywhere dense subset in $E$ has a cardinality at least equal to that of $B$, and that there exists an everywhere dense set in $E$ which is equipotent to $B$. b) Show that $\mathrm{Card}(E) = \mathrm{Card}(B^\mathbf{N})$ (to see that $\mathrm{Card}(E) \leq \mathrm{Card}(B^\mathbf{N})$ use a)). c) Show that if $\mathrm{Card}(B) \leq \mathrm{Card}(\mathbf{R})$, then the cardinality of every algebraic basis of $E$ is equal to $\mathrm{Card}(\mathbf{R}) = 2^{N_0}$ (use II, p. 80, exerc. 24, c)); however, if $\mathrm{Card}(B) > \mathrm{Card}(\mathbf{R})$, then every algebraic basis of $E$ is equipotent to $B^\mathbf{N}$ (use b) and A, II, § 7, exerc. 3, d)).