  The absolute logarithmic Weil height is well defined on the group of units of
the algebraic closure of the rational numbers, modulo roots of unity, and
induces a metric topology on this group. We show that the completion of this
metric space is a Banach space over the field of real numbers. We further show
that this Banach space is isometrically isomorphic to a co-dimension one
subspace of L1 of a certain totally disconnected, locally compact space,
equipped with a certain measure satisfying an invariance property with respect
to the absolute Galois group.
