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?
1-separated sequences of unit vectors in Banach spaces
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functional-analysis
banach-spaces
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1Would you settle for $\|u_k-u_l\|\geq 1-\varepsilon$? http://math.stackexchange.com/questions/4815/fitting-an-infinite-collection-of-balls-in-an-infinite-dimensional-unit-ball http://math.stackexchange.com/questions/163500/an-application-of-riesz-lemma – 2012-10-16
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5Elton 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 $n\ne m$. – 2012-10-16
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0@DavidMitra: Neat! It improves on Riesz's lemma, from which you can't generally get better than $1-\varepsilon$. – 2012-10-16
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0The 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. – 2012-10-16
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0See also http://math.stackexchange.com/q/296318/ – 2014-08-21