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柯西-施瓦茨不等式

柯西-施瓦茨不等式(英語:Cauchy–Schwarz inequality),又稱施瓦茨不等式柯西-布尼亞科夫斯基-施瓦茨不等式柯西不等式,在多個数学领域中均有應用的不等式;例如線性代數矢量數學分析無窮級數和乘積的積分,和概率論方差協方差。它被认为是最重要的数学不等式之一。它有一些推广,如赫尔德不等式

不等式以奧古斯丁·路易·柯西(Augustin Louis Cauchy),赫爾曼·阿曼杜斯·施瓦茨(Hermann Amandus Schwarz),和維克托·雅科夫列維奇·布尼亞科夫斯基英语Viktor_Bunyakovsky(Виктор Яковлевич Буняковский)命名。

叙述

是個複内积空间,則對所有的 有:

(a)
(b) 存在 使

證明請見内积空间#范数

特例

Rn-n维欧几里得空间

歐幾里得空間Rn,有

等式成立時:

也可以表示成

證明則須考慮一個關於的一個一元二次方程式

很明顯的,此方程式無實數或有重根,故其判別式

注意到

而等號成立於判別式

也就是此時方程式有重根,故

這兩例可更一般化赫爾德不等式

这是
n=3 时的特殊情况。

L2

对于平方可积复值函数的内积空间,有如下不等式:

赫尔德不等式是该式的推广。

矩阵不等式

列向量,则[a]

時不等式成立,设非零,,则
等号成立线性相关

Hermite阵,且,则

存在,设
等号成立线性相关

Hermite阵,且,则

存在,设
等号成立线性相关[1]

,则[2]

复变函数中的柯西不等式

在区域及其边界上解析,内一点,以为圆心做圆周 ,只要及其内部均被包含,则有:

其中,M是的最大值,

其它推广

[3]

[4]

參見

注释

  1. ^ 表示x的共轭转置

参考资料

  1. ^ 王松桂. 矩阵不等式-(第二版). 
  2. ^ 程伟丽 齐静. Cauchy不等式矩阵形式的推广. 郑州轻工业学院学报(自然科学版). 2008, (4) [2015-03-24]. (原始内容存档于2019-06-08). 
  3. ^ 赵明方. Cauchy不等式的推广. 四川师范大学学报(自然科学版). 1981, (2) [2015-03-24]. (原始内容存档于2019-06-03). 
  4. ^ 洪勇. 推广的Cauchy不等式的再推广. 曲靖师范学院学报. 1993, (S1) [2015-03-24]. (原始内容存档于2019-06-03). 

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