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Excerpt from Diagonal Scalings of the Laplacian as Preconditioners for Other Elliptic Differential Operators The spaces defined by and are both A-orthogonal and C-orthogonal to one another, and together they span all of R. The technique of expressing a given vector as a sum of vectors from each space will be used in the proofs throughout the paper. About the Publisher Forgotten Books publishes hundreds of thousands of rare and classic books. Find more at www.forgottenbooks.com This book is a reproduction of an important historical work. Forgotten Books uses state-of-the-art technology to digitally reconstruct the work, preserving the original format whilst repairing imperfections present in the aged copy. In rare cases, an imperfection in the original, such as a blemish or missing page, may be replicated in our edition. We do, however, repair the vast majority of imperfections successfully; any imperfections that remain are intentionally left to preserve the state of such historical works.