distances in $l^2$ This problem was posed by my friend and he said I may want to use some combinatorial set theory: Can you give me example of an uncountable $X \subseteq l^2$ (the Banach space of square summable sequences of reals) such that any two points in $X$ are at a rational distance?

Let $\sigma_n$ list all finite sequences of $0, 1$. Fix a bijection $f:\{
\sigma_n : n \geq 1\} \to \mathbb{N}$. Let $e_n = \langle 0, 0, \dots, 1, 0, 0, \dots \rangle \in l^2$ (1 occurs at the nth position). For each $x \subseteq \mathbb{N}$, let $a_x = \sum \left(\sqrt{3/2}\right)2^{-n} e_{f(x \upharpoonright n)}$. Now check.
