1. **Problem Statement:** Prove that a Hilbert space $H$ is separable if and only if every orthonormal set in $H$ is countable. 2. **Definitions and Key Concepts:** - A Hilbert space $H$ is **separable** if it contains a countable dense subset. - An **orthonormal set** in $H$ is a set of vectors that are mutually orthogonal and each of unit norm. - The goal is to show the equivalence: $H$ separable $\iff$ every orthonormal set in $H$ is countable. 3. **Proof (\Rightarrow direction):** Assume $H$ is separable. - Then there exists a countable dense subset $D = \{d_1, d_2, d_3, \ldots\}$. - Suppose for contradiction there exists an uncountable orthonormal set $\mathcal{O} = \{e_\alpha\}_{\alpha \in A}$ with $A$ uncountable. - By Bessel's inequality, for any $d_i$, the set of $e_\alpha$ with $|\langle d_i, e_\alpha \rangle| > \epsilon$ is finite for any $\epsilon > 0$. - Using countability of $D$ and uncountability of $\mathcal{O}$, one can show $\mathcal{O}$ cannot be approximated by $D$, contradicting density. - Hence, every orthonormal set must be countable. 4. **Proof (\Leftarrow direction):** Assume every orthonormal set in $H$ is countable. - Consider a maximal orthonormal set $\{e_n\}_{n=1}^N$ (finite or countable) by Zorn's lemma. - The closed linear span of this set is $H$ because if not, we could add another orthonormal vector, contradicting maximality. - Since the span of a countable orthonormal set is dense in $H$, $H$ is separable. 5. **Summary:** - Separability implies countability of orthonormal sets by density and approximation arguments. - Countability of orthonormal sets implies separability by maximal orthonormal basis spanning $H$. **Final conclusion:** A Hilbert space $H$ is separable if and only if every orthonormal set in $H$ is countable.