ε空间 I：实分析（第三年的数学博客选文）（英文版） [An Epsilon of Room,I:Real Analysis(Pages from year three of a Mathematic epub pdf mobi txt 电子书 下载 2024

ε空间 I：实分析（第三年的数学博客选文）（英文版） [An Epsilon of Room,I:Real Analysis(Pages from year three of a Mathematic epub pdf mobi txt 电子书 下载 2024

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ε空间 I：实分析（第三年的数学博客选文）（英文版） [An Epsilon of Room,I:Real Analysis(Pages from year three of a Mathematic epub pdf mobi txt 电子书 下载 2024

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内容简介

　　2007年，陶哲轩创立了一个内容丰富的数学博客，内容从他自己的研究工作和其他新近的数学进展，到他的授课讲义，包括各种非专业性难题和说明文章。头两年的博文已由美国数学会出版，而第三年的博文将分两册出版。第一册内容由实分析第二教程和博文中的相关资料构成。
　　实分析课程假定读者对一般测度论和本科分析的基本概念已有一定的了解。《ε空间 I：实分析（第三年的数学博客选文）（英文版）》内容包括：测度论中的高级专题，尤其是Lebesgue-Radon-Nikodym定理和Riesz表示定理；泛函分析专题，如Hilbert空间和Banach空间；广义函数空间和重要的函数空间，包括Lebesgue的Lp空间和Sobolev空间。另外还讨论了Fourier变换的一般理论。
　　《ε空间 I：实分析（第三年的数学博客选文）（英文版）》的第二部分谈到了许多辅助论题，诸如Zorn引理、Caratheodory延拓定理和Banach-Tarski悖论。作者还讨论了ε正规化推理——软分析的一个基本技巧，《ε空间 I：实分析（第三年的数学博客选文）（英文版）》书名正取于此意。总体来说，《ε空间 I：实分析（第三年的数学博客选文）（英文版）》提供了比二年级研究生实分析课程丰富得多的内容。
　　博文的第二册由各种专题的技术性和说明性文章组成，可以独立阅读。

内页插图

目录

Preface
A remark on notation
Acknowledgments
Chapter 1．Real analysis
1．1．A quick review of measure and integration theory
1．2．Signed measures and the Radon-Nikodym-Lebesgue theorem
1．3．Lp spaces
1．4．Hilbert spaces
1．5．Duality and the Hahn-Banach theorem
1．6．A quick review of point-set topology
1．7．The Baire category theorem and its Banach space consequences
1．8．Compactness in topological spaces
1．9．The strong and weak topologies
1．10．Continuous functions on locally compact Hausdorff spaces
1．11．Interpolation of Lp spaces
1．12．The Fourier transform
1．13．Distributions
1．14．Sobolev spaces
1．15．Hausdorff dimension
Chapter 2．Related articles
2．1．An alternate approach to the Caratheodory extension theorem
2．2．Amenability， the ping-pong lemma， and the Banach-
Tarski paradox
2．3．The Stone and Loomis-Sikorski representation theorems
2．4．Well-ordered sets， ordinals， and Zorn's lemma
2．5．Compactification and metrisation
2．6．Hardy's uncertainty principle
2．7．Create an epsilon of room
2．8．Amenability
Bibliography
Index

前言/序言

　　In February of 2007， I converted my "What's new" web page of research updates into a blog at terrytao .wordpress.com. This blog has since grown and evolved to cover a wide variety of mathematical topics， ranging from my own research updates， to lectures and guest posts by other mathematicians， to open problems， to class lecture notes， to expository articles at both basic and advanced levels.
　　With the encouragement of my blog readers， and also of the AMS， I published many of the mathematical articles from the first two years of the blog as [Ta2008] and [Ta2009]， which will henceforth be referred to as Structure and Randomn，ess and Poincare's Legacies Vols， I， H. This gave me the opportunity to improve and update these articles to a publishable (and citeable) standard， and also to record some of the substantive feedback I had received on these articles'by the readers of the blog.
　　The current text contains many (though not all) of the posts for the third year (2009) of the blog， focusing primarily on those posts of a mathematical nature which were not contributed primarily by other authors， and which are not published elsewhere. It has been split into two volumes.
　　The current volume consists oflecture notes from my graduate real anal- ysis courses that I taught at UCLA (Chapter 1)， together with some related material in Chapter 2. These notes cover the second part of the graduate real analysis sequence here， and therefore assume some familiarity with general measure theory (in particular， the construction of Lebesgue mea- sure and the Lebesgue integral， and more generally the material reviewed in Section 1.1)， as well as undergraduate real analysis (e.g.， various notions of limits and convergence). The notes then cover more advanced topics in measure theory (notably， the Lebesgue-Radon-Nikodym and Riesz representation theorems) as well as a number of topics in functional analysis， such as the theory of Hilbert and Banach spaces， and the study of key function spaces such as the Lebesgue and Sobolev spaces， or spaces of distributions.
　　The general theory of the Fourier transform is also discussed. In addition， a number of auxiliary (but optional) topics， such as Zorn's lemma， are discussed in Chapter 2. In my own course， I covered the material in Chapter 1 only and also used Folland's text [Fo2000] as a secondary source. But I hope that the current text may be useful in other graduate real analysis courses， particularly in conjunction with a secondary text (in particular， one that covers the prerequisite material on measure theory).
　　The second volume in this series (referred to henceforth as Volume H) consists of sundry articles on a variety of mathematical topics， which is onlyoccasionally related to the above course， and can be read independently.

ε空间 I：实分析（第三年的数学博客选文）（英文版） [An Epsilon of Room,I:Real Analysis(Pages from year three of a Mathematic epub pdf mobi txt 电子书 下载 2024

ε空间 I：实分析（第三年的数学博客选文）（英文版） [An Epsilon of Room,I:Real Analysis(Pages from year three of a Mathematic 下载 epub mobi pdf txt 电子书 2024

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