Functional Analysis: Introduction to Further Topics in Analysis (Princeton Lectures in Analysis, No. 4) (Hardcover)
暫譯: 函數分析:進一步分析主題介紹(普林斯頓分析講座,第4冊)
Elias M. Stein, Rami Shakarchi
- 出版商: Princeton University
- 出版日期: 2011-09-11
- 售價: $1,580
- 貴賓價: 9.8 折 $1,548
- 語言: 英文
- 頁數: 448
- 裝訂: Hardcover
- ISBN: 0691113874
- ISBN-13: 9780691113876
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商品描述
This is the fourth and final volume in "The Princeton Lectures in Analysis", a series of textbooks that aim to present, in an integrated manner, the core areas of analysis. Beginning with the basic facts of functional analysis, this volume looks at Banach spaces, Lp spaces, and distribution theory, and highlights their roles in harmonic analysis. The authors then use the Baire category theorem to illustrate several points, including the existence of Besicovitch sets. The second half of this book introduces readers to other central topics in analysis, such as probability theory and Brownian motion, which culminates in the solution of Dirichlet's problem. The concluding chapters explore several complex variables and oscillatory integrals in Fourier analysis, and illustrate applications to such diverse areas as nonlinear dispersion equations and the problem of counting lattice points. Throughout this book, the authors focus on key results in each area and stress the organic unity of the subject. This title offers a comprehensive and authoritative text that treats some of the main topics of modern analysis. It takes a look at basic functional analysis and its applications in harmonic analysis, probability theory, and several complex variables. It provides key results in each area that are discussed in relation to other areas of mathematics. It highlights the organic unity of large areas of analysis traditionally split into subfields. It provides interesting exercises and problems that illustrate ideas. Clear proofs provided.
商品描述(中文翻譯)
這是《普林斯頓分析講座》系列的第四卷也是最後一卷,這是一套旨在以整合的方式呈現分析核心領域的教科書。本卷從函數分析的基本事實開始,探討了巴拿赫空間(Banach spaces)、Lp 空間以及分佈理論,並強調它們在調和分析中的角色。作者接著使用貝爾類別定理(Baire category theorem)來說明幾個要點,包括 Besicovitch 集的存在性。本書的後半部分向讀者介紹分析中的其他核心主題,如概率論和布朗運動,最終解決了狄利克雷問題(Dirichlet's problem)。結尾章節探討了幾個複變數和傅立葉分析中的振盪積分,並展示了其在非線性色散方程和計數格點問題等多樣領域的應用。在整本書中,作者專注於每個領域的關鍵結果,並強調該主題的有機統一性。本書提供了一本全面且權威的文本,涵蓋現代分析的一些主要主題。它探討了基本的函數分析及其在調和分析、概率論和幾個複變數中的應用。它提供了每個領域的關鍵結果,並將其與數學的其他領域進行討論。它突顯了傳統上劃分為子領域的大範圍分析的有機統一性。書中還提供了有趣的練習和問題,以說明相關概念,並提供清晰的證明。