Abstract. This paper provides a framework for developing computationally efficient multilevel preconditioners and representations for Sobolev norms. Specifically, given a Hilbert space V and a nested sequence of subspaces V1  V2  . . .  V , we construct operators which are spectrally equivalent to those of the form A = k 							
						
							
					 															
					James H. Bramble, Joseph E. Pasciak, Panayot Vassi