歐氏空間集合級數在豪斯多夫距離下狄利克雷 審斂法

Euclidean Space Dirichlet’s test for Series of Sets Under Hausdorff Distance

許時耕(Shih-Keng Hsu)

指導教授 ： 吳裕振

中原大學/理學院/應用數學研究所/碩士(2012年)

https://doi.org/10.6840/cycu201200165

摘要

本篇論文主要對歐氏空間之集合級數之收斂性做研究,而此研究是把狄利克雷審斂法推廣到歐氏空間之集合級數,但狄利克雷審斂法是對 ΣAnBn 此型態做判斷其收斂性,因此我們必須定義集合的相加和相乘,尤其在相乘部份我們要滿足相乘後的集合必須還是在同一個維度,因此我們成功的把狄利克雷審斂法推廣到歐氏空間的集合級數,它是本論文的特色以及主要結果.

關鍵字

豪斯多夫距離 ； 狄利克雷審斂法 ； 集合級數

並列摘要

We mainly discuss the convergence of series of sets in Euclidean space in this paper, and we extend the Dirichlet’s test to the series of sets in Euclidean space. However, the Dirichlet’s test is used to judge the convergence
to the form of ΣAnBn. Thus, we must define the addition and the multiplication of series of sets, especially in the multiplication part, which is satisfied with the condition of the same dimension after multiplying. In consequence, it is successfully extended to the convergence of series of sets in Euclidean space, and which is the characteristic and the main result of the paper.

並列關鍵字

Series of sets ； Dirichlet’s test ； Hausdorff distance

參考文獻

[1] Apostol, T. M., (1974). Mathematical Analysis, 2nd edition, Addison-Wesley, Reading, Massachusetts.

[2] C. Castaing and M. Valadiner, (1997). Convex Analysis and Measurable Multifunctions, Lecture Note in Math 580, springer-verlag.

[3] Taylor, A. E. and Lay, D. C., (1980). Introduction to Functional Analysis, 2nd edition, Wiley New York.

[4] 李俊霖, (2012). Euclidean Space Absolutely Convergent Series of Sets Under Hausdorff Distance, 中原大學, 碩士論文.

參考文獻