第 1 页摘 要Cauchy-Schwarz 不等式是数学中重要的不等式之一，在较多的不同领域中应用广泛。本文所研究的是 Cauchy-Schwarz 不等式在不同的数学的领域中几种常见的不同的基本形式及其证明方法，并对 Cauchy-Schwarz 不等式的推广作了一些系统的论述。在此基础上，本文分别给出了柯西-施瓦茨不等式在概率论与数理统计和机器学习中的应用，在许多问题中起到良好的效果。在文章的最后，本文还给出了柯西-施瓦茨不等式的更一般的形式，即著名的赫尔德不等式并给出了相应的应用。关键词: 柯西-施瓦茨不等式， 概率论与数理统计，机器学习，赫尔德不等式第 2 页AbstractCauchy-Schwartz Inequality is one of the most important inequalities in mathematical analysis, which is widely used in many different fields. In this paper, several common expressions of Cauchy -Schwartz Inequality are summarized, and the corresponding proofs are given, and the generalization of Cauchy-Schwartz Inequality is systematically discussed. On this basis, this paper presents the application of Cauchy -Schwartz Inequality in probability statistics and machine learning. At the end of the paper, the more general form of Cauchy -Schwartz Inequality, namely, Hölder Inequality, and its application are given.Keyword Cauchy-Schwarz inequality, probability theory and statistics, machine learning, Holder inequality.第 3 页目录摘 要.................................................................................................................................2Abstract..............................................................................................................................3第一章 引言符号解释............................................................................................................51.1 引言.......................................................................................................................51.2 符号解释.......................................................................................
