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Given an infinite-dimensional Banach space $X$, I would like to construct a sequence of linearly independent unit vectors such that $\|u_k-u_l\|\geqslant 1$ whenever $k\neq l$. Any ideas on how to realize this?

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Would you settle for $\|u_k-u_l\|\geq 1-\varepsilon$?…… – Jonas Meyer Oct 16 '12 at 20:42
Elton and Odell proved that in an infinite dimensional normed linear space, there is an $\epsilon>0$ and a sequence $(x_n)$ of unit vectors that satisfy $\Vert x_n-x_m\Vert\ge 1+\epsilon$ for $n\ne m$. This (difficult result) can be found in the last chapter of Joseph Diestel's Sequences and Series in Banach Spaces. In chapter one of the same book, it is shown, fairly simply (and attributed to Cliff Kottman), that in an infinite dimensional normed space, one can find a sequence $(x_n)$ satisfying $\Vert x_n-x_m\Vert>1$ for $n\ne m$. – David Mitra Oct 16 '12 at 20:52
@DavidMitra: Neat! It improves on Riesz's lemma, from which you can't generally get better than $1-\varepsilon$. – Jonas Meyer Oct 16 '12 at 21:03
The original paper of Kottman is: Kottman, C. A. 1975. Subsets of the unit ball that are separated by more than one. Studia math., 53, 15-27. A simpler proof of the result is in Diestel's book, and is attributed to Tom Starbird. – David Mitra Oct 16 '12 at 21:05
See also – Jonas Meyer Aug 21 '14 at 23:27

It can be done fairly easily in any infinite dimensional normed space $X$. For a proof, see Lemma 1.4.22 in Robert E. Megginson's An Introduction to Banach Space Theory.

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