  Let $X$ be an infinite dimensional uniformly smooth Banach space. We prove
that $X$ contains an infinite equilateral set. That is, there exists a constant
$\lambda>0$ and an infinite sequence $(x_i)_{i=1}^\infty\subset X$ such that
$\|x_i-x_j\|=\lambda$ for all $i\neq j$.

Functional Analysis