Kadets, M.I. ; Kadets, V.M. Series in Banach spaces. Conditional and unconditional convergence / Birkhauser, , 1997 (Operator Theory Advances and Applications, vol. 94), 156 pp.Известно, что для бесконечных сумм коммутативный закон уже не верен: сумма может меняться при перестановке слагаемых. Монография посвящена вопросам, связанным с этим эффектом. Содержание:
Chapter 1.
Background Material (Numerical Series. Riemann's Theorem, Main Definitions. Elementary Properties of Vector Series, Preliminary material on Rearrangements of Series of Elements of a Banach Space).
Chapter 2.
Series in a Finite-Dimensional Space (Steinitz's Theorem on the Sum Range of a Series, The Dvoretzky-Hanani Theorem on Perfectly Divergent Series, Pecherskii's Theorem).
Chapter 3.
Conditional Convergence in an Infinite-Dimensional Space (Basic Counterexamples, A Series Whose Sum Range Consists of Two Points, Chobanyan's Theorem, The Khinchin Inequalities and the Theorem of M. I. Kadets on Conditionally Convergent Series in L_p).
Chapter 4.
Unconditionally Convergent Series (The Dvoretzky-Rogers Theorem, Orlicz's Theorem on Unconditionally Convergent Series in L_p, Absolutely Summing Operators. Grothendieck's Theorem).
Chapter 5.
Orlicz's Theorem and the Structure of Finite-Dimensional Subspaces (Finite Representability, The space c_o, C-Convexity, and Orlicz's Theorem, Survey on Results on Type and Cotype).
Chapter 6.
Some Results from the General Theory of Banach Spaces (Frechet Differentiability of Convex Functions, Dvoretzky's Theorem, Basic Sequences, Some Applications to Conditionally Convergent Series).
Chapter 7.
Steinitz's Theorem and B-Convexity (Conditionally Convergent Series in Spaces with Infratype, A Technique for Transferring Examples with Nonlinear Sum Range to Arbitrary Infinite-Dimensional Hanach Spaces, Series in Spaces That Are Not B-Convex).
Chapter 8.
Rearrangements of Series in Topological Vector Spaces (Weak and Strong Sum Range, Rearrangements of Series of Functions, Banaszczyk's Theorem on Series in Metrizable Nuclear Spaces).
Appendix. The Limit Set of the Riemann Integral Sums of a Vector-Valued Function
