# Spooky geometry IV

Let $[a,b]$ and $[c,d]$ be subintervals of $[0,1]$ and call them non-overlapping if they intersect only possibly in their endpoints. Construct a continuous curve $\sigma:[0,1]\to H$ with $H$ a Hilbert space, such that if $[a,b]$ and $[c,d]$ are any non-overlapping subintervals of $[0,1]$, then $\sigma(b)-\sigma(a)$ and $\sigma(d)-\sigma(c)$ are orthogonal.

Harder: show that this can be carried out in any infinite-dimensional Hilbert space.