TITLE: 1-separated sequences of unit vectors in Banach spaces
QUESTION [5 upvotes]: 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?

REPLY [2 votes]: 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.