In this paper, a key problem of the rigorous formulation of the
renormalization group as a continuous flow is identified. Some essential
features of the operator-theoretic renormalization group are recalled, and a
family of norms associated to different Banach spaces of Hamiltonians is
introduced. From this, three different properties, which should be satisfied by
an adequate norm, are derived, e.g., a submultiplicativity with respect to
Wick-ordering. A proof is given that none of the norms introduced in this paper
satisfy all of these conditions.