  We define an inner product on a vector space of adelic measures over a number
field. We find that the norm induced by this inner product governs weak
convergence at each place of $K$. The canonical adelic measure associated to a
rational map is in this vector space, and the square of the norm of the
difference of two such adelic measures is the Arakelov-Zhang pairing from
arithmetic dynamics. We prove a sharp lower bound on the norm of adelic
measures with points of small adelic height. We find that the norm of a
canonical adelic measure associated to a rational map is commensurate with the
Arakelov height on the space of rational functions with fixed degree. As a
consequence, the Arakelov-Zhang pairing of two rational maps $f$ and $g$ can be
bounded from below as a function of $g$.
