PSEUDOCOMPLEMENTATION IN BANACH-SPACES

This paper introduces the following definition: a closed subspace Z of a Banach space E is pseudocomplemented in E if for every linear continuous operator u from Z to Z there is a linear continuous extension uBAR of u from E to E. For instance, every subspace complemented in E is pseudocomplemented in E. First, the pseudocomplemented hilbertian subspaces of L1 are characterized and, in L(p) with p in [1, + infinity[, classes of closed subspaces in which the notions of complementation and pseudocomplementation are equivalent are pointed out. Then, for Banach spaces with the uniform approximation property, Dvoretzky's theorem is strengthened by proving that they contain uniformly pseudocomplemented l(n)2's. Finally, the study of Banach spaces in which every closed subspace is pseudocomplemented is started.