The Korteweg-de Vries equation (KdV) @tv(x;t )+ @ 3 xv (x;t) 3@xv(x;t) 2 =0 ( x2 S 1 ;t2R ) is a completely integrable system with phase space L 2 (S 1 ). Although the Hamiltonian H(q ): = R S 1 1 @ x q ( x ) 2 + q ( x ) 3 dx is dened only on the dense subspace H 1 (S 1 ), we prove that the frequencies !j = @H @Jj can be dened on the whole space L 2 (S 1 ), where (Jj)j1 denote the action variables which are globally dened on L 2 (S 1 ). These frequencies are real analytic functionals and can be used to analyze Bourgain’s weak solutions of KdV with initial data in L 2 (S 1 ). The same method can be used for any equation in the KdV hierarchy.