[proofplan] For existence, we take a countable dense subset (from separability), extract a linearly independent subcollection with the same closure, apply [Gram-Schmidt](/theorems/542), and verify completeness via the density criterion from the [Characterisation of Complete Orthonormal Systems](/theorems/541). For countability, we use the $\sqrt{2}$-separation of orthonormal elements to inject any orthonormal system into a countable dense subset. [/proofplan] [step:Extract a linearly independent set whose span is dense] Let $\{w_j\}_{j=1}^\infty$ be a countable dense subset of $H$. Extract a linearly independent subcollection $\{v_k\}$ inductively by discarding elements already in the span of those retained. [claim:Dense Span Preservation] $\overline{\operatorname{span}\{v_k\}} = H$. [/claim] [proof] Every discarded $w_j$ lies in $\operatorname{span}\{v_k\}$, and every retained $w_j$ is some $v_k$. So $\{w_j\} \subseteq \operatorname{span}\{v_k\}$, and density gives $\overline{\operatorname{span}\{v_k\}} = H$. [/proof] [/step] [step:Apply Gram-Schmidt and verify completeness] Apply [Gram-Schmidt Orthonormalisation](/theorems/542) to $\{v_k\}$ to obtain an orthonormal system $\{e_k\}$ with $\overline{\operatorname{span}\{e_k\}} = \overline{\operatorname{span}\{v_k\}} = H$. By the [Characterisation of Complete Orthonormal Systems](/theorems/541) (condition 2), this density implies $\{e_k\}$ is complete. [/step] [step:Show every orthonormal system is at most countable via the $\sqrt{2}$-separation argument] Let $\{f_\alpha\}_{\alpha \in A}$ be an orthonormal system. For $\alpha \neq \beta$, $\|f_\alpha - f_\beta\|_H^2 = 2$ by the Pythagorean theorem. The open balls $B(f_\alpha, 1/2)$ are pairwise disjoint. Since $H$ is separable, every collection of pairwise disjoint open balls is at most countable (each contains a distinct element of a countable dense subset). Therefore $A$ is at most countable. [/step]