Theorem. - All orthonormal bases of an Hilbert space $H$ have the same cardinality. It follows that the concept of dimension of a Hilbert space is well-defined.

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Proof: When $H$ is finite-dimensional (as a vector space), every orthonormal basis is a Hamel basis of $H$ . Thus, the result follows from the fact that all Hamel bases of a vector space have the same cardinality (see this entry).

We now consider the case where $H$ is infinite-dimensional (as a vector space). Let $\{e_i\}_{i \in I}$ and $\{f_j\}_{j \in J}$ be two orthonormal basis of $H$ , indexed by the sets $I$ and $J$ , respectively. Since $H$ is infinite dimensional the sets $I$ and $J$ must be infinite.

We know, from Parseval's equality, that for every $x \in H$

Hence, since each $I_j$ is countable, $I \leq J\!\times\!\mathbb{N} \cong J$ (because $J$ is infinite).

An analogous argument proves that $J \leq I$ . Hence, by the Schroeder-Bernstein theorem $J$ and $I$ have the same cardinality. $\square$