Non-separable Hilbert space

Question

Let $H$ be a Hilbert space.

Suppose that $H$ has an uncountable orthonormal set.

Why is $H$ is non-separable?

Answer 1

Let $S$ be an uncountable orhonormal set. For each $v\in S$, consider the open ball $B_{1/2}(v)$. Then $v\neq w\implies B_{1/2}(v)\cap B_{1/2}(w)=\emptyset$. Therefore, if $D$ is a dense subset of $H$, $D$ must have at least one element in each such ball and, since distinct balls are disjoint and there are uncountably many of them, $D$ must be uncountable. So, no countable subset of $H$ can possibly be dense.