Sequences and Series in Banach Spaces

1. Front Matter

   Pages iii-xii

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2. Riesz’s Lemma and Compactness in Banach Spaces

   • Joseph Diestel

   Pages 1-8

3. The Weak and Weak* Topologies: An Introduction

   • Joseph Diestel

   Pages 9-16

4. The Eberlein-Šmulian Theorem

   • Joseph Diestel

   Pages 17-23

5. The Orlicz-Pettis Theorem

   • Joseph Diestel

   Pages 24-31

6. Basic Sequences

   • Joseph Diestel

   Pages 32-57

7. The Dvoretsky-Rogers Theorem

   • Joseph Diestel

   Pages 58-65

8. The Classical Banach Spaces

   • Joseph Diestel

   Pages 66-123

9. Weak Convergence and Unconditionally Convergent Series in Uniformly Convex Spaces

   • Joseph Diestel

   Pages 124-146

10. Extremal Tests for Weak Convergence of Sequences and Series

    • Joseph Diestel

    Pages 147-172

11. Grothendieck’s Inequality and the Grothendieck-Lindenstrauss-Pelczynski Cycle of Ideas

    • Joseph Diestel

    Pages 173-191

12. An Intermission: Ramsey’s Theorem

    • Joseph Diestel

    Pages 192-199

13. Rosenthal’s l ₁ Theorem

    • Joseph Diestel

    Pages 200-218

14. The Josefson-Nissenzweig Theorem

    • Joseph Diestel

    Pages 219-225

15. Banach Spaces with Weak Sequentially Compact Dual Balls

    • Joseph Diestel

    Pages 226-240

16. The Elton-Odell (1 + ε)-Separation Theorem

    • Joseph Diestel

    Pages 241-255

17. Back Matter

    Pages 257-263

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This volume presents answers to some natural questions of a general analytic character that arise in the theory of Banach spaces. I believe that altogether too many of the results presented herein are unknown to the active abstract analysts, and this is not as it should be. Banach space theory has much to offer the prac­ titioners of analysis; unfortunately, some of the general principles that motivate the theory and make accessible many of its stunning achievements are couched in the technical jargon of the area, thereby making it unapproachable to one unwilling to spend considerable time and effort in deciphering the jargon. With this in mind, I have concentrated on presenting what I believe are basic phenomena in Banach spaces that any analyst can appreciate, enjoy, and perhaps even use. The topics covered have at least one serious omission: the beautiful and powerful theory of type and cotype. To be quite frank, I could not say what I wanted to say about this subject without increasing the length of the text by at least 75 percent. Even then, the words would not have done as much good as the advice to seek out the rich Seminaire Maurey-Schwartz lecture notes, wherein the theory's development can be traced from its conception. Again, the treasured volumes of Lindenstrauss and Tzafriri also present much of the theory of type and cotype and are must reading for those really interested in Banach space theory.

• Banach
• Banach Space
• Banachscher Raum
• Convexity
• Sequences
• Series
• Spaces
• choquet integral
• compactness
• differential equation
• extrema

• Department of Math Sciences, Kent State University, Kent, USA

  Joseph Diestel

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