  Concentration compactness method is a powerful techniques for establishing
existence of minimizers for inequalities and of critical points of functionals
in general. The paper gives a functional-analytic formulation for the method in
Banach space, generalizing the Hilbert space case elaborated in \cite{ccbook}.
The key object is a dislocation space - a triple $(X,F,D)$, where $F$ is a
convex functional that defines a norm on Banach space $X$, and $D$ is a group
of isometries on $X$. Bounded sequences in dislocation spaces admit a
decomposition into an asymptotic sum "profiles" $w^{(n)}\in X$ dislocated by
actions of $D$, that is, a sum of the form $\sum_ng^{(n)}_kw^{(n)}$,
$g^{(n)}_k\in D$, while the remainder term converges weakly under actions of
any sequence $g_k\in D$ ({\em $D$-weak convergence}). This decomposition allows
to extend the weak convergence argument from variational problems with
compactness to problems where $X$ is {\em cocompactly} (relatively to the group
$D$) imbedded into a Banach space $Y$, that is, when every sequence $D$-weakly
convergent in $X$ is convergent in the norm of $Y$. We prove a general
statement on existence of minimizers in cocompact imbeddings that applies, in
particular to Sobolev imbeddings which lack compactness (unbounded domain,
critical exponent) including the subelliptic Sobolev spaces and spaces over
Riemannian manifolds.
