Real Mathematical Analysis (Undergraduate Texts in Mathematics)

Charles Chapman Pugh

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商品描述

Based on an honors course taught by the author at UC Berkeley, this introduction to undergraduate real analysis gives a different emphasis by stressing the importance of pictures and hard problems. Topics include: a natural construction of the real numbers, four-dimensional visualization, basic point-set topology, function spaces, multivariable calculus via differential forms (leading to a simple proof of the Brouwer Fixed Point Theorem), and a pictorial treatment of Lebesgue theory. Over 150 detailed illustrations elucidate abstract concepts and salient points in proofs. The exposition is informal and relaxed, with many helpful asides, examples, some jokes, and occasional comments from mathematicians, such as Littlewood, Dieudonné, and Osserman. This book thus succeeds in being more comprehensive, more comprehensible, and more enjoyable, than standard introductions to analysis.

New to the second edition of Real Mathematical Analysis is a presentation of Lebesgue integration done almost entirely using the undergraph approach of Burkill. Payoffs include: concise picture proofs of the Monotone and Dominated Convergence Theorems, a one-line/one-picture proof of Fubini's theorem from Cavalieri’s Principle, and, in many cases, the ability to see an integral result from measure theory. The presentation includes Vitali’s Covering Lemma, density points ― which are rarely treated in books at this level ― and the almost everywhere differentiability of monotone functions. Several new exercises now join a collection of over 500 exercises that pose interesting challenges and introduce special topics to the student keen on mastering this beautiful subject.

商品描述(中文翻譯)

本書是根據作者在加州大學伯克利分校所教授的榮譽課程而寫的,介紹了大學生實分析的不同重點,強調圖像和難題的重要性。主題包括:實數的自然構造、四維視覺化、基礎點集拓撲學、函數空間、通過微分形式進行的多變量微積分(從而簡單證明了布劳威尔不動點定理),以及對勒贝格理論的圖像化處理。超過150個詳細的插圖闡明了抽象概念和證明中的重點。本書的表達方式非正式輕鬆,並附有許多有用的旁白、例子、一些笑話,以及數學家(如利特伍德、迪厄多內和奧瑟曼)的偶爾評論。因此,本書比標準的分析入門書更全面、更易理解、更有趣。



第二版《實數分析》的新內容包括幾乎完全使用Burkill的下圖法進行的勒貝格積分介紹。其中的收益包括:單調收斂定理和被支配收斂定理的簡潔圖像證明,從騎士原理對Fubini定理的一行/一張圖的證明,以及在許多情況下,從測度論中看到積分結果的能力。該介紹還包括維塔利覆蓋引理、密度點(在這個級別的書籍中很少涉及)以及單調函數的幾乎處處可微性。現在新增了幾個新練習題,與超過500個練習題一起,為那些渴望掌握這一美妙學科的學生提供了有趣的挑戰和特殊主題的介紹。