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 $D = \{x_1, x_2, \ldots\}$ in $H$. - Suppose for contradiction there exists an uncountable orthonormal set $\mathcal{O} = \{e_\alpha\}_{\alpha \in A}$ with $A$ uncountable. - For any two distinct vectors $e_\alpha, e_\beta$ in $\mathcal{O}$, $\|e_\alpha - e_\beta\|^2 = \|e_\alpha\|^2 + \|e_\beta\|^2 = 2$ because they are orthonormal. - This means the vectors in $\mathcal{O}$ are at least $\sqrt{2}$ apart. - Since $D$ is dense, for each $e_\alpha$ there exists $d_\alpha \in D$ with $\|e_\alpha - d_\alpha\| < \frac{\sqrt{2}}{2}$. - But the balls of radius $\frac{\sqrt{2}}{2}$ around each $e_\alpha$ are disjoint due to the distance between $e_\alpha$'s. - This implies an injection from the uncountable set $A$ into the countable set $D$, a contradiction. - 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) in $H$. - By the Hilbert space theory, the closed linear span of this set is $H$. - Since the set is countable, the linear combinations with rational coefficients form a countable dense subset. - Therefore, $H$ is separable. 5. **Summary:** - If $H$ is separable, no uncountable orthonormal set can exist. - If every orthonormal set is countable, $H$ has a countable orthonormal basis, hence is separable. **Final answer:** A Hilbert space $H$ is separable if and only if every orthonormal set in $H$ is countable.