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所在平台: Udemy |
课程主页: https://www.udemy.com/course/real-analysis-part-2/
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**课程名称：** 实分析（二）：函数序列与级数 **课程概述：** 本课程是实分析系列课程的第二部分，深入探讨实数、实数序列与级数以及实变函数的行为。课程重点关注实值序列和函数的几个关键性质，包括收敛性、极限、连续性、光滑性、可微性和可积性。 与研究复数及其函数的复分析不同，实分析专注于实数系统的分析。本课程将从序列与级数的基本概念入手，涵盖子序列、柯西序列（并探讨如波尔查诺-魏尔斯特拉斯定理、黎曼定理等重要定理），以及序列的上、下极限。此外，课程还将深入讨论实数和复数级数的重排以及相关的定理。 特别地，课程将详细讲解序列和子序列的各项概念，并着重探讨了与连续性相关的议题，包括连续函数、一致连续性，以及连续性与紧致性之间的联系。 实分析中的许多概念可以从实直线推广到更广泛、更抽象的数学背景。这种推广使得实分析与其他数学分支，如拓扑学（通过将连续函数和紧致性概念推广到度量空间和拓扑空间）以及泛函分析（通过将有限维欧几里得空间推广到无限维空间，从而引出巴拿赫空间和希尔伯特空间等概念）产生联系。 **学习支持与成果：** 学生将在 instructor 的支持下完成课程学习，并通过在线考试获得课程结业证书。证书将通过一个独特的链接进行追踪。
In mathematics, the branch of real analysis studies the behavior of real numbers, sequences and series of real numbers, and real functions.[1] Some particular properties of real-valued sequences and functions that real analysis studies include convergence, limits, continuity, smoothness, differentiability and integrability.Real analysis is distinguished from complex analysis, which deals with the study of complex numbers and their functions.Sequence and series is a content from Real Analysis which mugs up the Basic concept of sequences and series , Subsequences , Cauchy Sequence with expected theorems like Bolzano Weierstrass Theorem , Riemann Theorem , Concept of Upper and Lower Limit of a sequence , Rearrangement of series of Real and Complex numbers. An Interesting and Expected Theorems on: Sequences and subsequence with detailed Concepts including_ Continuity Continuous Functions Uniform Continuity Continuity and CompactnessVarious ideas from real analysis can be generalized from the real line to broader or more abstract contexts. These generalizations link real analysis to other disciplines and subdisciplines. For instance, generalization of ideas like continuous functions and compactness from real analysis to metric spaces and topological spaces connects real analysis to the field of general topology, while generalization of finite-dimensional Euclidean spaces to infinite-dimensional analogs led to the concepts of Banach spaces and Hilbert spaces and, more generally to functional analysis.The student will support for their queries from the instructor and get a certificate of completion in the end of the course which can also be tracked via unique link.
