Series in Banach Spaces

• 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.

“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)