effective Hilbertspace for the functional dependence of the truncated set of parameters.\
We introduce the blocking concept discussed in the previous section as a tensor product decomposition of the Hilbert space $$\label{grad1}
{\cal H}\,=\,\underset{I\in {\frak I}}{\bigotimes}\, {\cal H}_{I}\,,$$ where ${\frak I}$ denotes some indexing set for the blocks. We are looking for an embedding and truncation map which respects the block decomposition by factorization $$\begin{aligned}
\label{grad2}
G_{{\cal H}'}\,=\,\underset{I\in {\frak I}} {\bigotimes}\, G_{{\cal H}'_{I}}
\quad\mbox{and}\quad
G^+_{{\cal H}}\,=\,\underset{I\in {\frak I}} {\bigotimes}\, G^+_{{\cal H}_{I}}\,.\end{aligned}$$ Using this mathematical formulation of the blocking scheme we like to write the RG transformation for a block in an analogous way $$\label{grad3}
{\cal O}_{{\cal H}'_{I}}\left({\bf K}'\right)
\,=\, G^+_{{\cal H}_{I}}\,\circ\,{{\cal O}_{{\cal H}_{I}}\left({\bf
      K}\right)}\,
  \circ\,G_{{\cal H}'_{I}}\;,$$ due to (\[grad2\]). But equation (\[grad3\]) is not an independent relation since we have to relate it to the [*global*]{} relation (\[transf4\]). To decompose (\[transf4\]) into the blocked pieces (\[grad3\]) we have to assume that the operator ${\cal O}_{\cal H}$ can be decomposed into commuting block operators ${\cal O}_{{\cal H}_{I}}$ which is not the case in general in quantum physics. Therefore the problem encountered so far is to find suitable functions ${\cal O}_{\cal H}\left({\bf K}\right)$ which respect the block decomposition of the Hilbert space 