Schauder Bases in Quotients of Subspaces of l 2 (X)

Abstract

Introduction. Questions and problems related to Schauder basis are among the most classical ones in the Banach space theory and have been studied from the earliest days. The fact that every infinite-dimensional Banach space has an infinite-dimensional subspace with a basis, has been observed already in Banach's book [Ba]. At about the same time, the existence of bases in classical Banach spaces, such as C(0,1) and Lp(0,1), was established. An importance of the concept of a basis lies in the fact that it provides a natural method of approximation of vectors and operators in the space. A question related to the approximation property in Banach spaces was asked by Mazur in The Scottish Book [Scb]. This question and the problem of the existence of a basis in every Banach space stayed open for about 40 years, although not because of the lack of interest (cf. e.g. [Sin]). In 1972 P Enflo constructed a Banach space which fails the approximation property and hence does not admit a Schauder basis. His discovery triggered a flurry of activities resulting in constructions of subspaces failing the approximation property in all classical Banach spaces not isomorphic to a Hilbert space. Such subspaces have been also found in all Banach spaces sufficiently far from the Hilbert space (cf. e.g. [L-T.l], [L-T.2] and references therein). Techniques developed until that moment did not provide quantitative finitedimensional estimates for basis constants. It took another 10 years before a new probabilistic argument was introduced by E. D. Gluskin [G.l], and consequently the existence of n-dimensional Banach spaces with basis constant tending to infinity was proved independently by Gluskin [G.2] and S. J. Szarek [Sz.l]. At last a circle got closed: a method of constructing infinite-dimensional examples out of finite-dimensional ones was proposed by J. Bourgain [B.l] and developed by Szarek [Sz.3], to obtain an /2-sum of finite-dimensional Banach spaces which has no basis, but obviously admits a so-called finite-dimensional decomposition. In this paper we prove a striking isomorphic characterization of a Hilbert space in terms of a Schauder basis. First observe that the class of Banach spaces isomorphic to a Hilbert space is closed under operations of passing to closed