Series in Banach Spaces

“The material in the book is always started with motivation. There are plenty of exercises, with hints for solution. The list of references is complete and up-to-day. This is an amazing book on doctoral level, written by experienced authors, who contributed a lot to the subject presented in the book.” (J.Musielak, zbMATH, 0876.46009, 1997)

• Notations.- 1. Background Material.- §1. Numerical Series. Riemann’s Theorem.- §2. Main Definitions. Elementary Properties of Vector Series.- §3. Preliminary Material on Rearrangements of Series of Elements of a Banach Space.- 2. Series in a Finite-Dimensional Space.- §1. Steinitz’s Theorem on the Sum Range of a Series.- §2. The Dvoretzky-Hanani Theorem on Perfectly Divergent Series.- §3. Pecherskii’s Theorem.- 3. Conditional Convergence in an Infinite-Dimensional Space.- §1. Basic Counterexamples.- §2. A Series Whose Sum Range Consists of Two Points.- §3. Chobanyan’s Theorem.- §4. The Khinchin Inequalities and the Theorem of M. I. Kadets on Conditionally Convergent Series in Lp.- 4. Unconditionally Convergent Series.- §1. The Dvoretzky-Rogers Theorem.- §2. Orlicz’s Theorem on Unconditionally Convergent Series in LpSpaces.- §3. Absolutely Summing Operators. Grothendieck’s Theorem.- 5. Orlicz’s Theorem and the Structure of Finite-Dimensional Subspaces.- §1. Finite Representability.- §2.The space c0, C-Convexity, and Orlicz’s Theorem.- §3. Survey on Results on Type and Cotype.- 6. Some Results from the General Theory of Banach Spaces.- §1. Fréchet Differentiability of Convex Functions.- §2. Dvoretzky’s Theorem.- §3. Basic Sequences.- §4. Some Applications to Conditionally Convergent Series.- 7. Steinitz’s Theorem and B-Convexity.- §1. Conditionally Convergent Series in Spaces with Infratype.- §2. A Technique for Transferring Examples with Nonlinear Sum Range to Arbitrary Infinite-Dimensional Banach Spaces.- §3. Series in Spaces That Are Not B-Convex.- 8. Rearrangements of Series in Topological Vector Spaces.- §1. Weak and Strong Sum Range.- §2. Rearrangements of Series of Functions.- §3. Banaszczyk’s Theorem on Series in Metrizable Nuclear Spaces.- Appendix. The Limit Set of the Riemann Integral Sums of a Vector-Valued Function.- §2. The Example of Nakamura and Amemiya.- §4. Connection with the Weak Topology.- Comments to the Exercises.- References.