Asymptotic infinite-dimensional theory of Banach spaces

Abstract

In this paper structure of infinite dimensional Banach spaces is studied by using an asymptotic approach based on stabilization at infinity of finite dimensional subspaces which appear everywhere far away. This leads to notions of asymptotic structures and asymptotic versions of a given Banach space. As an example of application of this approach, a class of asymptotic $l_p$-spaces is introduced and investigated in detail. Some properties of this class, as duality and complementation, are analogous to properties of classical $l_p$ spaces, although the latter is more ``regular'' than its classical counterpart; in contrast, the property exhibited in the uniqueness theorem is very different than for spaces $l_p$.