Bases in Banach Spaces I - Ivan Singer

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Sommaire:
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. SpecialClasses of Bases in Banach Spaces.- I. Classes of Bases not Involving Unconditional Convergence.- II. Unconditional Bases and Some Classes of Unconditional Bases.- Notation Index.- Author Index.