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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.

bryanj
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    $\gamma$ is defined on $[0,1]$, no? But anyway, you do need an infinite dimension. A vocabulary question: “one-to-one“ means injective, or bijective? – Aphelli Jan 26 '19 at 23:25
  • Thanks - in Rudin's vocabulary it means injective. – bryanj Jan 27 '19 at 01:08

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