The Banach Space over the Field of Real Numbers

A Banach space determined by the Weil height

We show that 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, modulo roots of unity, andinduces 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 onesubspace of l1 of a certain totally disconnected, locally compact space equipped with a certain measure satisfying an invariance property with respectto the absolute galois group.