  We consider pairs of Banach spaces (M_0, M) such that M_0 is defined in terms
of a little-o condition, and M is defined by the corresponding big-O condition.
The construction is general and pairs include function spaces of vanishing and
bounded mean oscillation, vanishing weighted and weighted spaces of functions
or their derivatives, M\"obius invariant spaces of analytic functions,
Lipschitz-H\"older spaces, etc. It has previously been shown that the bidual
M_0** of M_0 is isometrically isomorphic with M. The main result of this paper
is that M_0 is an M-ideal in M. This has several useful consequences: M_0 has
Pelczynskis properties (u) and (V), M_0 is proximinal in M, and M_0* is a
strongly unique predual of M, while M_0 itself never is a strongly unique
predual.
