  For many known non-compact embeddings of two Banach spaces $E\hookrightarrow
F$, every bounded sequence in $E$ has a subsequence that takes form of a
profile decomposition - a sum of clearly structured terms with asymptotically
disjoint supports plus a remainder that vanishes in the norm of $F$. In this
note we construct a profile decomposition for arbitrary sequences in the
Sobolev space $H^{1,2}(M)$ of a compact Riemannian manifold, relative to the
embedding of $H^{1,2}(M)$ into $L^{2^*}(M)$, generalizing the well-known
profile decomposition of Struwe ([Proposition 2.1]{Struwe}) to the case of
arbitrary bounded sequences.
