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        應用泛函分析(第1卷)

        出版時間:2009-10  出版社:世界圖書出版公司  作者:澤德勒  頁數:481  
        Tag標簽:無  

        前言

          More precisely, by (i), I mean a systematic presentation of the materialgoverned by the desire for mathematical perfection and completeness ofthe results. In contrast to (i), approach (ii) starts out from the question"What are the most important applications?" and then tries to answer thisquestion as quickly as possible. Here, one walks directly on the main roadand does not wander into all the nice and interesting side roads.  The present book is based on the second approach. It is addressed toundergraduate and beginning graduate students of mathematics, physics,and engineering who want to learn how functional analysis elegantly solvesma hematical problems that are related to our real world azld that haveplayed an important role in the history of mathematics. The reader shouldsense that the theory is being developed, not simply for its own sake, butfor the effective solution of concrete problems.

        內容概要

          More precisely, by (i), I mean a systematic presentation of the materialgoverned by the desire for mathematical perfection and completeness ofthe results. In contrast to (i), approach (ii) starts out from the question"What are the most important applications?" and then tries to answer thisquestion as quickly as possible. Here, one walks directly on the main roadand does not wander into all the nice and interesting side roads.  The present book is based on the second approach. It is addressed toundergraduate and beginning graduate students of mathematics, physics,and engineering who want to learn how functional analysis elegantly solvesma hematical problems that are related to our real world azld that haveplayed an important role in the history of mathematics. The reader shouldsense that the theory is being developed, not simply for its own sake, butfor the effective solution of concrete problems.

        書籍目錄

        PrefacePrologueContents of AMS Volume 1091  Banach Spaces and Fixed-Point Theorems  1.1  Linear Spaces and Dimension  1.2  Normed Spaces and Convergence  1.3  Banach Spaces and the Cauchy Convergence Criterion  1.4  Open and Closed Sets  1.5  Operators  1.6  The Banach Fixed-Point Theorem and the Iteration Method  1.7  Applications to Integral Equations  1.8  Applications to Ordinary Differential Equations  1.9  Continuity  1.10 Convexity  1.11 Compactness  1.12 Finite-Dimensional Banach Spaces and Equivalent Norms  1.13 The Minkowski Functional and Homeomorphisms  1.14 The Brouwer Fixed-Point Theorem  1.15 The Schauder Fixed-Point Theorem  1.16 Applications to Integral Equations  1.17 Applications to Ordinary Differential Equations  1.18 The Leray-Schauder Principle and a priori Estimates  1.19 Sub-and Supersolutions, and the Iteration Method in Ordered Banach Spaces  1.20 Linear Operators  1.21 The Dual Space  1.22 Infinite Series in Normed Spaces  1.23 Banach Algebras and Operator Functions  1.24 Applications to Linear Differential Equations in Banach Spaces  1.25 Applications to the Spectrum  1.26 Density and Approximation  1.27 Summary of Important Notions2  Hilbert Spaces, Orthogonality, and the Dirichlet  Principle  2.1  Hilbert Spaces  2.2  Standard Examples  2.3  Bilinear Forms  2.4  The Main Theorem on Quadratic Variational Problems  2.5  The Functional Analytic Justification of the Dirichlet Principle  2.6  The Convergence of the Ritz Method for Quadratic Variational Problems  2.7  Applications to Boundary-Value Problems, the Method of Finite Elements, and Elasticity  2.8  Generalized Functions and Linear Functionals  2.9  Orthogonal Projection  2.10 Linear Functionals and the Riesz Theorem  2.11 The Duality Map  2.12 Duality for Quadratic Variational Problems  2.13 The Linear Orthogonality Principle  2.14 Nonlinear Monotone Operators  2.15 Applications to the Nonlinear Lax-Milgram Theorem and the Nonlinear Orthogonality Principle3  Hilbert Spaces and Generalized Fourier Series  3.1  Orthonormal Series  3.2  Applications to Classical Fourier Series  3.3  The Schmidt Orthogonalization Method  3.4  Applications to Polynomials  3.5  Unitary Operators  3.6  The Extension Principle  3.7  Applications to the Fourier Transformation  3.8  The Fourier Transform of Tempered Generalized Functions4  Eigenvalue Problems for Linear Compact Symmetric  Operators  ……5  Self-Adjoint Operators, the Friedrichs Extension and the Partial Differential Equations of Mathematical physicsEpilogueAppendixReferencesHints for Further ReadingList of SymbolsList of TheoremsList of the Most Important DefinitionsSubject Index

        章節摘錄

          I think that time is ripe for such an approach. From a general point of view,functional analysis is based on an assimilation of analysis, geometry, alge-bra, and topology. The applications to be considered concern the followingtopics: ordinary differential equations (initial-value problems, boundary-eigen-value problems, and bifurcation); linear and nonlinear integral equations;variational problems, partial differential equations, and Sobolev spaces;optimization (e.g., Cebyev approximation, control of rockets, game the-ory, and dual problems);Fourier series and generalized Fourier series;the Fourier transformation,generalized functions (distributions) and the role of the Green function;partial differential equations of mathematical physics (e.g., the Laplaceequation, the heat equation, the wave equation, and the Schr6dinger equa-tion);time evolution and semigroups;the N-body problem in celestial mechanics;capillary surfaces;minimal surfaces and harmonic maps;superfluids, superconductors, and phase transition (the Landau-Ginz-burg model).

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        用戶評論 (總計5條)

         
         

        •   雖然已經有幾本經典的泛函分析專著,但這本對于我這樣的非數學系的人來說,可能更合適,就買了本。
        •   如果能仔細讀下來,你將發現這是非常好的泛函分析的書。語言生動,理論詳細而系統,例子經典,淺顯易懂。所有定理,推論都有證明。特別講清了泛函分析的概念與古典的數學分析中的聯系,因為泛函分析的有些概念就是古典概念的在新的情況下的推廣。我是在自己33歲以后才自學泛函分析,但憑這本書,我已經非常喜歡基礎數學了!我覺得還有幾本書讀起來就像是在讀經典小說那樣引人入勝,愛不釋手。比如:黎茨的《泛函分析講義》,羅伊登的《實分析》等等國外的作品。
        •   泛函的入門書,很實用的說。
        •   不錯,好好讀讀。
        •   這是第一卷,還有第二卷!兩本都買了!講的很詳實!
         

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