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About this product
- Author(s)Ivan Singer
- PublisherSpringer-Verlag Berlin and Heidelberg GmbH & Co. KG
- Date of Publication14/05/2012
- Series TitleGrundlehren der mathematischen Wissenschaften
- Series Part/Volume Number154
- Place of PublicationBerlin
- Country of PublicationGermany
- ImprintSpringer-Verlag Berlin and Heidelberg GmbH & Co. K
- Content NoteVIII, 668 p.
- Weight987 g
- Edition StatementSoftcover reprint of the original 1st ed. 1970
- Table Of ContentsI. 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.
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