Before revision:  A necessary and sufficient condition for integration of order d=1,2,\dots is given in terms of the decomposition of the space H into the direct sum of d+1 closed subspaces \tau_h, h=0, , d, each one associated with components of the process integrated of order h. These results mirror the ones recently obtained in the finite dimensional case, with the only difference that the number of cointegrating relations  of order 0 is infinite  .
After revision:  A necessary and sufficient condition for integration of order d=1,2,\dots is given in terms of the decomposition of the space H into the direct sum of d+1 closed subspaces \tau_h, h=0, , d, each one associated with components of the process . Except for the fact that the number of cointegrating relations of order 0 is infinite .