ON M-IDEALS AND o-O TYPE SPACES

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, Mobius invariant spaces of analytic functions, Lipschitz-Holder 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.