Theorem 1 (Stone)

The mapping $\sigma$ which to a unitary representation $\varrho$ of $\mathbf{R}$ in E associates its infinitesimal generator $u$ defines a bijection of the set of unitary representations of $\mathbf{R}$ in E onto the set of self-adjoint partial operators on E. The reciprocal bijection $\tau$ associates to a self-adjoint partial operator the unitary representation $t\mapsto e^{itu}$.

Let us first prove some lemmas.