應(yīng)用泛函分析（第1卷）

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(nèi)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(jié)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）.

圖書封面

圖書標(biāo)簽Tags

無(wú)

評(píng)論、評(píng)分、閱讀與下載

---

    應(yīng)用泛函分析（第1卷） PDF格式下載

---