By the Stone–von Neumann theorem there is a unitary operator U from L (R) onto F , unique up to multiplication by a scalar, intertwining the two representations of the Weyl commutation relations. By Schur's lemma and the Gelfand–Naimark construction, the matrix coefficient of any vector determines the vector up to a scalar multiple. Since the matrix coefficients of F = E 0 and f = H 0 are equal, it follows that the unitary U is uniquely determined by the properties