Let

1(1)={x=(xi)i1:xi0,  i=1xi=1}.\ell_1^{(1)} = \left\{ x=(x_i)_{i\ge1} : x_i\ge 0,\; \sum_{i=1}^{\infty} x_i =1 \right\}.

Equip 1(1)\ell_1^{(1)} with the metric induced by the 1\ell^1 norm:

d(x,y)=xy1=...d(x,y) = \|x-y\|_1 = ...