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內容簡介

　　2007年，陶哲軒創立瞭一個內容豐富的數學博客，內容從他自己的研究工作和其他新近的數學進展，到他的授課講義，包括各種非專業性難題和說明文章。頭兩年的博文已由美國數學會齣版，而第三年的博文將分兩冊齣版。第一冊內容由實分析第二教程和博文中的相關資料構成。
　　實分析課程假定讀者對一般測度論和本科分析的基本概念已有一定的瞭解。《ε空間 I：實分析（第三年的數學博客選文）（英文版）》內容包括：測度論中的高級專題，尤其是Lebesgue-Radon-Nikodym定理和Riesz錶示定理；泛函分析專題，如Hilbert空間和Banach空間；廣義函數空間和重要的函數空間，包括Lebesgue的Lp空間和Sobolev空間。另外還討論瞭Fourier變換的一般理論。
　　《ε空間 I：實分析（第三年的數學博客選文）（英文版）》的第二部分談到瞭許多輔助論題，諸如Zorn引理、Caratheodory延拓定理和Banach-Tarski悖論。作者還討論瞭ε正規化推理——軟分析的一個基本技巧，《ε空間 I：實分析（第三年的數學博客選文）（英文版）》書名正取於此意。總體來說，《ε空間 I：實分析（第三年的數學博客選文）（英文版）》提供瞭比二年級研究生實分析課程豐富得多的內容。
　　博文的第二冊由各種專題的技術性和說明性文章組成，可以獨立閱讀。

內頁插圖

目錄

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空間 I：實分析（第三年的數學博客選文）（英文版） [An Epsilon of Room,I:Real Analysis(Pages from year three of a Mathematic 下載 mobi epub pdf txt 電子書

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