IMC 2008 · Problem 6

Let $\mathcal{H}$ be an infinite-dimensional real Hilbert space, let $d > 0$, and suppose that $S$ is a set of points (not necessarily countable) in $\mathcal{H}$ such that the distance between any two distinct points in $S$ is equal to $d$. Show that there is a point $y \in \mathcal{H}$ such that

$$\left\{ \frac{\sqrt{2}}{d}(x - y) \;:\; x \in S \right\}$$

is an orthonormal system of vectors in $\mathcal{H}$.