On bases in Banach spaces

Abstract

We investigate various kinds of bases in infinite dimensional Banach spaces. In particular, we consider the complexity of Hamel bases in separable and non-separable Banach spaces and show that in a separable Banach space a Hamel basis cannot be analytic, whereas there are non-separable Hilbert spaces which have a discrete and closed Hamel basis. Further we investigate the existence of certain complete minimal systems in `∞ as well as in separable Banach spaces. Outline. The paper is concerned with bases in infinite dimensional Banach spaces. The first section contains the definitions of the various kinds of bases and biorthogonal systems and also summarizes some set-theoretic terminology and notation which will be used throughout the paper. The second section provides a survey of known or elementary results. The third section deals with Hamel bases and contains some consistency results proved using the forcing technique. The fourth section is devoted to complete minimal systems (including Φ-bases and Auerbach bases) and the last section contains open problems. ∗The research for this paper began during the Workshop on Set Theory, Topology, and Banach Space Theory, which took place in June 2003 at Queen’s University Belfast, whose hospitality is gratefully acknowledged. The workshop was supported by the Nuffield Foundation Grant NAL/00513/G of the third author, the EPSRC Advanced Fellowship of the second author and the grant GACR 201/03/0933 of the fourth author. 1