Orlicz-Pettis型定理的相关问题

One of the most interesting and useful theorems in the early history of functional analysis is a result now known as the Orlicz-Pettis Theorem. Many scholars have been devoted to the study of conclusions on Orlicz-Pettis Theorem since the first one was given. . In recent years, the concepts of statistical convergence and almost convergence are introduced to establish Orlicz-Pettis type theorems. These concepts are somewhat complicated, which result in many open problems to be solved. We will establish and improve some conclusions on such problems to develop the theories. For example, we will establish a version of Hahn-Schur theorem for statistical convergence and statistical Cesaro summing; establish a version of Orlicz-Pettis theorem for statistical convergence on locally convex space; establish a version of Orlicz-Pettis theorem for statistical Cesaro convergence and bounded multiplier convergence; discuss whether some classical conclusions are true or not if weak convergence is replace by weak Cesaro convergence.

Orlicz-Pettis型定理是泛函分析的历史中最具趣味性和实用性的定理之一. 自该定理建立起至今, 许多学者致力于对其相关理论的研究.. 近些年来, 统计收敛和几乎收敛被引入了 Orlicz-Petti型定理的建立. 但是由于其定义的复杂性, 关于统计收敛和几乎收敛的 Orlicz-Petis型定理的相关结论还有很多尚未成型. 为了完善这一理论, 我们将建立和改进一系列关于统计收敛和几乎收敛的相关结论. 例如, 建立关于统计收敛和统计 Cesaro求和的 Hahn-Shur型结论; 在局部凸空间中建立关于统计收敛的 Orlicz-Pettis型定理; 建立关于统计 Cesaro求和的 Orlicz-Pettis型定理的有界乘数收敛版本; 在经典的关于弱收敛的结论中, 讨论将弱收敛替换成弱 Cesaro收敛的可行性.

Orlicz-Pettis型定理是泛函分析的历史中最具趣味性和实用性的定理之一. 自该定理建立起至今, 许多学者致力于对其相关理论的研究. .我们在局部凸空间中建立关于统计收敛的Orlicz-Pettis型定理; 讨论了弱K收敛和弱cesaro收敛点列的性质。