Bases in Banach Spaces II
Published on 1. February 1981
Book
Hardback
VIII, 880 pages
978-3-540-10394-3 (ISBN)
Description
Since the appearance, in 1970, of Vol. I of the present monograph 1370], the theory of bases in Banach spaces has developed substantially. Therefore, the present volume contains only Ch. III of the monograph, instead of Ch. Ill, IV and V, as was planned initially (cp. the table of contents of Vol. I). Since this volume is a continuation of Vol. I of the same monograph, we shall refer to the results of Vol. I directly as results of Ch. I or Ch. II (without specifying Vol. I). On the other hand, sometimes we shall also mention that certain results will be considered in Vol. III (Ch. IV, V). In spite of the many new advances made in this field, the statement in the Preface to Vol. I, that "the existing books on functional analysis contain only a few results on bases", remains still valid, with the exception of the recent book [248 a] of J. Lindenstrauss and L. Tzafriri. Since we have learned about [248 a] only in 1978, in this volume there are only references to previous works, instead of [248 a]; however, this will cause no inconvenience, since the intersec tion of the present volume with [248 a] is very small. Let us also mention the appearance, since 1970, of some survey papers on bases in Banach spaces (V. D. Milman [287], [288], C. W. McArthur [275]; M. I. Kadec [204], § 3 and others).
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Content
Vol. II. Chapter III. Generalizations of the Notion of a Basis.- § 0. Banach spaces which do not have the approximation property.- I. Countable Generalizations of Bases.- § 1. Basic sequences. Bibasic systems.- § 2. Deficient basic sequences. Images and inverse images of bases by continuous linear mappings.- § 3. Complete sequences.- § 4. Bases with respect to a class of sequences of indices.- § 5. Pseudo-bases. Semi-bases.- § 6. Minimal sequences. {?i}-linearly independent sequences. Complete minimal sequences. Maximal biorthogonal systems.- § 7. Generalized bases.- § 8. M-bases. Strong M-bases. Series summable M-bases.- § 9. Approximative bases. Quasi-bases. Finite-dimensional expansions of the identity. Commuting approximative bases.- §10. Operational bases. Generalized summation bases.- § 11. T-bases (summation bases). Strongly series summable M-bases. ?-bases.- § 12. ?-bases. ?-bases. ?1?-bases. The universal complements Cp.- § 13. Dual ?-bases. Commuting ?-bases. Bases with parentheses. Finite-dimensional decompositions.- § 14. Duality theorems. Further universal complement properties of the spaces Cp.- § 15. Decompositions (bases of subspaces). Schauder decompositions. Resolutions of the identity. Integral bases.- § 16. Bases of sets.- II. Generalizations of Bases, Without Assuming Countability.- § 17. ER-sets. Extended unconditional bases. Transfinite bases. Strongly unconditional integral bases.- § 18. Extended approximative bases. Extended ?-bases. Extended resolutions of the identity.- § 19. Transfinite decompositions. Transfinite Schauder decompositions. Ordinal resolutions of the identity.- § 20. Extended biorthogonal systems. Extended M-bases.- Notes and Remarks.- Appendix: Complements added in proof.- Bibliography to the Appendix.- Notation Index.- Author Index.