Bases in Banach Spaces I

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.