Asymptotic L P Spaces and Bounded Distortions

The new class of Banach spaces, so-called asymptotic lp spaces, is introduced and it is shown that every Banach space with bounded distortions contains a subspace from this class. The proof is based on an investigation of certain functions, called enveloping functions, which are intimately connected with stabilization properties of the norm. 0 Introduction During the last year several problems of infinite-dimensional Banach space theory, which remained open for decades, have been finally solved. Some new constructions of Banach spaces have been made which, on one hand, showed limitations of the theory, but on the other hand, also showed how exciting an infinite-dimensional geometry can be. Let us mention few of them: (i) a space without unconditional basic sequence (Gowers–Maurey), (ii) a space not isomorphic to any of its hyperplanes (Gowers), (iii) a space such that every bounded operator being a Fredholm operator (Gowers–Maurey). The problems which were answered by these examples are of a lineartopological nature. Although a thorough study of this kind of properties flourished back in the 60s, methods developed that time and later were not sufficient to succesfully atack these problems. The solutions given last year are by-products of a study in a different direction: the infinite-dimensional