Bases in Banach Spaces I

This monograph attempts to present the results known today on bases in Banach spaces and some unsolved problems concerning them. Although this important part of the theory of Banach spaces has been studied for more than forty years by numerous mathematicians, the existing books on functional analysis (e. g. M. M. Day [43], A. Wilansky [263], R. E. Edwards [54]) contain only a few results on bases. A survey of the theory of bases in Banach spaces, up to 1963, has been presented in the expository papers [241], [242] and [243], which contain no proofs; although in the meantime the theory has rapidly deve1oped, much of the present monograph is based on those expository papers. Independently, a useful bibliography of papers on bases, up to 1963, was compiled by B. L. Sanders [219J. Due to the vastness of the field, the monograph is divided into two volumes, ofwhich this is the first (see the tab1e of contents). Some results and problems re1ated to those treated herein have been de1iberately planned to be inc1uded in Volume 11, where they will appear in their natural framework (see [242], [243]).

I. The Basis Problem. Some Properties of Bases in Banach Spaces.- § 1. Definition of a basis in a Banach space. The basis problem. Relations between bases in complex and real Banach spaces.- §2. Some examples of bases in concrete Banach spaces. Some separable Banach spaces in which no basis is known.- § 3. The coefficient functional associated to a basis. Bounded bases. Normalized bases.- §4. Biorthogonal systems. The partial sum operators. Some characterizations of regular biorthogonal systems. Applications.- § 5. Some characterizations of regular E-complete biorthogonal systems. Multipliers.- § 6. Some types of linear independence of sequences.- § 7. Intrinsic characterizations of bases. The norm and the index of a sequence. The index of a Banach space. Extension of block basic sequences.- § 8. Domination and equivalence of sequences. Equivalent, affinely equivalent and permutatively equivalent bases.- § 9. Stability theorems of Paley-Wiener type.- § 10. Other stability theorems.- §11. An application to the basis problem.- § 12. Properties of strong duality. Application : bases and sequence spaces.- § 13. Bases in topological linear spaces. Weak bases and bounded weak bases in Banach spaces. Weak* bases and bounded weak* bases in conjugate Banach spaces.- § 14. Schauder bases in topological linear spaces. Properties of weak duality for bases in Banach spaces.- § 15. (e)-Schauder bases and (b)-Schauder bases in topological linear spaces.- § 16. Some remarks on bases in normed linear spaces.- §17. Continuous linear operators in Banach spaces with bases.- §18. Bases of tensor products.- § 19. Best approximation in Banach spaces with bases.- § 20. Polynomial bases. Strict polynomial bases. ? systems and ? systems.- Notes and remarks.- II. Special Classes of Bases in Banach Spaces.- I. Classes of Bases not Involving Unconditional Convergence.- § 1. Monotone and strictly monotone bases.- § 2. Normal bases.- § 3. Positive bases.- § 4. k-shrinking bases.- § 5. Retro-bases in conjugate Banach spaces.- § 6. k-boundedly complete bases.- § 7. Bases of types wc0, (wc0)*, swc0 and (swc0)*.- § 8. Some properties of the set of all elements of a basis. Weakly closed and (weakly closed)* bases.- § 9. Bases of types P, P*, aP and aP*.- § 10. Bases of types l+, (l+)*, al+ and (al+)*. The cone associated to a basis.- § 11. Besselian and Hilbertian bases. Stability theorems.- § 12. Relations between various types of bases.- § 13. Universal bases. Complementably universal bases. Block-universal bases.- II. Unconditional Bases and Some Classes of Unconditional Bases.- § 14. Unconditional bases. Conditional bases.- § 15. Some separable Banach spaces having no unconditional basis.- § 16. Some characterizations of unconditional bases among E-complete (or total) biorthogonal systems and among bases. Some characterizations by properties of the associated cone. Multipliers.- § 17. Intrinsic characterizations of unconditional bases. Some more separable Banach spaces having no unconditional basis. Properties of strong duality. Unconditional bases and sequence spaces.- § 18. Equivalence and permutative equivalence of unconditional bases. Universal unconditional bases.- § 19. Best approximation in Banach spaces with unconditional bases.- § 20. Orthogonal bases. Strictly orthogonal bases. Hyperorthogonal and strictly hyperorthogonal bases.- § 21. Subsymmetric bases.- § 22. Symmetric bases. Symmetric spaces.- § 23. Applications: Existence of non-equivalent normalized bases and conditional bases in infinite dimensional Banach spaces with bases.- § 24. Perfectly homogeneous bases. Application: Banach spaces with a unique normalized unconditional basis.- § 25. Absolutely convergent bases. Uniform bases.- Notes and remarks.- Notation Index.- Author Index.