§2. 一致收敛函数列与函数项级数的性质

掌握一致收敛函数列的极限函数的连续性、可积性、可微性定理。掌握一致收敛函数项级数的连续性、逐项积分、逐项求导定理。

重点习题：例2、例3

 

 

迪尼定理(Dini theorem)是一个分析方向的数学定理，提出者是意大利数学家、政治家乌利塞·迪尼(Ulisse Dini)，是关于实值函数序列的一致收敛性判定的定理。

 

{f_n}是闭区间[a,b]上一连续实值函数列且满足下述两个条件：

 

(1) {f_n}逐点收敛到一个连续函数f，

 

(2) {f_n}在每个点上都是单调序列，

 

那么{f_n}在[a,b]上一致收敛到f.

 

 

 

 

 

Ulisse Dini (14 November 1845-28 October 1918) was an Italian mathematician and politician, born in Pisa. He is known for his contribution to real analysis, partly collected in his book "Fondamenti per la teorica delle funzioni di variabili reali".

 

Dini attended the Scuola Normale Superiore in order to become a teacher. One of his professors was Enrico Betti. In 1865, a scholarship enabled him to visit Paris, where he studied under Charles Hermite as well as Joseph Bertrand, and published several papers. In 1866, he was appointed to the University of Pisa, where he taught algebra and geodesy. In 1871, he succeeded Betti as professor for analysis and geometry. From 1888 until 1890, Dini was rettore of the Pisa University, and of the Scuola Normale Superiore from 1908 until his death in 1918.

 

He was also active as a politician: in 1871 he was voted into the Pisa city council, and in 1880, he became a member of the Italian parliament.