Proposition 7

In a hilbertian space E, two orthonormal bases are equipotent.

Let B and C be two orthonormal bases of E. The case when one of the two sets B, C is finite is trivial, since a finite orthonormal basis is an algebraic basis of the space. Suppose therefore that B and C are infinite. For every $x \in B$, let $C_x$ be the subset of C consisting of all $y \in C$ such that $\langle x | y \rangle \neq 0$. The set $C_x$ is countable (V, p. 21, prop. 4). For every $y \in C$, there exists $x \in B$ such that $y \in C_x$, since B is an orthonormal basis and $y \neq 0$; in other words C is the union of the countable sets $C_x$ as $x$ ranges over B. The cardinality of C is hence less than that of $\mathbf{N} \times B$, hence less than that of B (S, III, § 6, No. 3, cor. 4); similarly, the cardinality of B is less than that of C; this completes the proof.

The cardinality of an arbitrary orthonormal basis of a hilbertian space E is called the hilbertian dimension of E.