Abstract/Summary

We consider pairs of Banach spaces (M0,M)(M0,M) such that M0M0 is defined in terms of a little-oo condition, and MM is defined by the corresponding big-OO 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, Möbius invariant spaces of analytic functions, Lipschitz-Hölder spaces, etc. It has previously been shown that the bidual M∗∗0M0∗∗ of M0M0 is isometrically isomorphic with MM. The main result of this paper is that M0M0 is an M-ideal in MM. This has several useful consequences: M0M0 has Pełczýnskis properties (u) and (V), M0M0 is proximinal in MM, and M∗0M0∗ is a strongly unique predual of MM, while M0M0 itself never is a strongly unique predual.