# Concentration compactness

The **concentration compactness** principle of Lions is a way of compensating for the well known failure of precompactness of bounded
sets in infinite-dimensional Banach spaces (i.e. that bounded sequences need not have convergent subsequences). The principle, roughly speaking, asserts that given any bounded sequence, there exists a subsequence which resolves into the superposition of convergent sequences that have been shifted by an asymptotically orthogonal set of unitary group actions, plus an error term which goes to zero in certain coarse norms which are weaker than the original norm topology (but significantly stronger than the weak topology).

It is useful in generating nonlinear profiles of solutions to nonlinear equations, and also combines well with the induction on energy method.

Concentration compactness has similar goals as compensated compactness, but achieves them by different means.