Approximation properties and non-linear geometry of Banach spaces

Abstract. The existence of Lipschitz quasiadditive projections on linear subspaces is investigated. A new characterization of the bounded approx-imation property provides an alternative proof of the equivalence between this property and its Lipschitz version. We show that Lipschitz-free spaces over finite-dimensional normed spaces have a finite-dimensional decomposi-tion. The purpose of this work is to investigate the approximation properties of the natural preduals of spaces of Lipschitz functions on metric spaces. Such preduals are sometimes called free spaces and although their definition is fairly straightforward, their structure remains largely unknown. In the first section, we investigate in a fairly general context Lipschitz quasiadditive projections (or equivalently, Lipschitz liftings). Special attention is given to M-ideals, and it is shown in particular that when a non-reflexive space is an M-ideal in its bidual, no decent quasiadditive projection exists (Proposition 1.6), which answers negatively a question from [7, p. 56]. Ando’s theorem [2, Theorem 5] shows however that M-ideals in separable spaces lead to a quite different situation, and when combined with [6, Cor. 3.2] in section 2, it provides a characterization of the bounded approximation property 2000 Mathematics Subject Classification. 46B20; 46B28; 54H05. Key words and phrases. Approximation properties, Lipschitz-free space, quasiadditiv