Author | Kadets, Mikhail I. author |
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Title | Series in Banach Spaces [electronic resource] : Conditional and Unconditional Convergence / by Mikhail I. Kadets, Vladimir M. Kadets |
Imprint | Basel : Birkhรคuser Basel, 1997 |
Connect to | http://dx.doi.org/10.1007/978-3-0348-9196-7 |
Descript | VIII, 159 p. online resource |
Notations -- 1. Background Material -- ยง1. Numerical Series. Riemannโs Theorem -- ยง2. Main Definitions. Elementary Properties of Vector Series -- ยง3. Preliminary Material on Rearrangements of Series of Elements of a Banach Space -- 2. Series in a Finite-Dimensional Space -- ยง1. Steinitzโs Theorem on the Sum Range of a Series -- ยง2. The Dvoretzky-Hanani Theorem on Perfectly Divergent Series -- ยง3. Pecherskiiโs Theorem -- 3. Conditional Convergence in an Infinite-Dimensional Space -- ยง1. Basic Counterexamples -- ยง2. A Series Whose Sum Range Consists of Two Points -- ยง3. Chobanyanโs Theorem -- ยง4. The Khinchin Inequalities and the Theorem of M. I. Kadets on Conditionally Convergent Series in Lp -- 4. Unconditionally Convergent Series -- ยง1. The Dvoretzky-Rogers Theorem -- ยง2. Orliczโs Theorem on Unconditionally Convergent Series in LpSpaces -- ยง3. Absolutely Summing Operators. Grothendieckโs Theorem -- 5. Orliczโs Theorem and the Structure of Finite-Dimensional Subspaces -- ยง1. Finite Representability -- ยง2. The space c0, C-Convexity, and Orliczโs Theorem -- ยง3. Survey on Results on Type and Cotype -- 6. Some Results from the General Theory of Banach Spaces -- ยง1. Frรฉchet Differentiability of Convex Functions -- ยง2. Dvoretzkyโs Theorem -- ยง3. Basic Sequences -- ยง4. Some Applications to Conditionally Convergent Series -- 7. Steinitzโs Theorem and B-Convexity -- ยง1. Conditionally Convergent Series in Spaces with Infratype -- ยง2. A Technique for Transferring Examples with Nonlinear Sum Range to Arbitrary Infinite-Dimensional Banach Spaces -- ยง3. Series in Spaces That Are Not B-Convex -- 8. Rearrangements of Series in Topological Vector Spaces -- ยง1. Weak and Strong Sum Range -- ยง2. Rearrangements of Series of Functions -- ยง3. Banaszczykโs Theorem on Series in Metrizable Nuclear Spaces -- Appendix. The Limit Set of the Riemann Integral Sums of a Vector-Valued Function -- ยง2. The Example of Nakamura and Amemiya -- ยง4. Connection with the Weak Topology -- Comments to the Exercises -- References