  The concept of a profile decomposition formalizes concentration compactness
arguments on the functional-analytic level, providing a powerful refinement of
the Banach-Alaoglu weak-star compactness theorem. We prove existence of profile
decompositions for general bounded sequences in uniformly convex Banach spaces
equipped with a group of bijective isometries, thus generalizing analogous
results previously obtained for Sobolev spaces and for Hilbert spaces. Profile
decompositions in uniformly convex Banach spaces are based on the notion of
$\Delta$-convergence by T. C. Lim instead of weak convergence, and the two
modes coincide if and only if the norm satisfies the well-known Opial
condition, in particular, in Hilbert spaces and $\ell^{p}$-spaces, but not in
$L^{p}(\mathbb R^{N})$, $p\neq2$. $\Delta$-convergence appears naturally in the
context of fixed point theory for non-expansive maps. The paper also studies
connection of $\Delta$-convergence with Brezis-Lieb Lemma and gives a version
of the latter without an assumption of convergence a.e.
