dependence given by the Hamiltonian, we can construct $G^+$ as a projection map from the space of all eigenvectors of the Hamiltonian to a space containing a reduced number of eigenvectors. A projection map from this truncated space back to the space containing all eigenvectors is a natural way of defining $G$. Although such an example illustrates a possible application of the abstract formulation given by (\[transf4\]) it raises the question of which eigenstates are necessary to keep for constructing the truncated space. In the case of zero temperature we can argue that only those eigenvectors should be kept, which correspond to the low energy eigenvalues [@bl82]. As pointed out previously the idea of this paper should be to invent a real-space RG formulation which overcomes these limitations by a more abstract formulation.\
Let us now assume that the functional dependence ${\cal O}$ is given by some operator, not necessarily the Hamiltonian, on the original Hilbertspace ${\cal H}$ so that equation (\[transf4\]) can be written as the commuting diagram $$\label{comm1}
\begin{CD}
{\cal H}' @> G >> {\cal H}\\
@V{\cal O}\left({\bf K}'\right) VV  @VV{\cal O}\left({\bf K}\right) V\\
{\cal H}' @<< G^+ < {\cal H}
\end{CD}$$ where ${\cal H}'$ refers to the 