The absolute logarithmic Weil height is well defined on the quotient group Q/ Tor ( Q ) and induces a metric topology in this group. We show that the completion of this metric space is a Banach space over the field R of real numbers. We further show that this Banach space is isometrically isomorphic to a co-dimension one subspace of L1(Y,B, λ), where Y is a certain totally disconnected, locally compact space, B is the σ-algebra of Borel subsets of Y , and λ is a certain measure satisfying an… CONTINUE READING