I want to do a sanity check on this exercise, which is stated in the book as
Show [given a Hilbert space $H$] that there is a continuous one-to-one mapping $\gamma$ of $[0, 1]$ into $H$ such that $\gamma(b) - \gamma(a)$ is orthogonal to $\gamma(d) - \gamma(c)$ whenever $0 \le a \le b \le c \le d \le 1$.
He does not specify anything about the dimension of $H$, but it seems to me that there be a countably infinite orthonormal set in $H$ for this to be true.
Take $u_n = \gamma(2^n) - \gamma(2^{n-1})$.
Am I thinking straight here?
Thanks.