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Concentration compactness is an important method in mathematical analysis which has been widely used in mathematical research for two decades. This unique volume fulfills the need for a sourcebook that usefully combines a concise formulation of the method, a range of important applications, and background material concerning manifolds, non-compact transformations and functional spaces. Highlighting the role in functional analysis of invariance and, in particular, of non-compact transformation groups, the book uses the same building blocks, such as partitions of domain and partitions of range, relative to transformation groups, in the proofs of energy inequalities and in the weak convergence lemmas.