On the Hilbert space derived from the Weil distribution

We study the Hilbert space obtained by completing the space of all smooth and compactly supported functions on the real line with respect to the hermitian form arising from the Weil distribution under the Riemann hypothesis. It turns out that this Hilbert space is isomorphic to a de Branges space by a composition of the Fourier transform and a simple map.This result is applied to state a new equivalence condition for the Riemann hypothesis in a series of equalities.