图书信息

作者简介

内容简介

目录

图书信息

出版社: 高等教育出版社; 第1版 (2007年2月1日)

平装: 580页

正文语种: 英语

开本: 16

ISBN: 7040216493

条形码: 9787040216493

尺寸: 22.6 x 16.8 x 2.8 cm

重量: 721 g

作者简介

作者：(美国)拉克斯

内容简介

《泛函分析(影印版)》是美国科学院院士Peter D．Lax在CotJrant数学所长期讲授泛函分析课程的教学经验基础上编写的。《泛函分析(影印版)》包括泛函分析的基本内容：Barlach空间、Hilbert空间和线性拓扑空间的基本概念和性质，线性拓扑空间中的凸集及其端点集的性质，有界线性算子的性质等。可作为本科生泛函分析课的教学内容；还包括泛函分析较深的内容：自伴算子的谱分解理论。紧算子的理论，交换Barlach代数的Gelfand理论，不变子空间的理论等。可作为研究生泛函分析课的教学内容。《泛函分析(影印版)》特别强调泛函分析与其他数学分支的联系及泛函分析理论的应用，可以使读者深刻地理解到：抽象的泛函分析理论有着丰富的数学背景。

目录

Foreword

1. Linear Spaces

Axioms for linear spaces-Infinite-dimensional examples-Subspace, linear span-Quotient space-Isomorphism-Convex sets-Extreme subsets

2. Linear Maps

2.1 Algebra of linear maps,

Axioms for linear maps-Sums and composites-Invertible linear maps-Nullspace and range-Invariant subspaces

2.2. Index of a linear map,

Degenerate maps-Pseudoinverse-IndexmProduct formula for the index-Stability of the index

3. The Hahn,Banach Theorem

3.1 The extension theorem,

Positive homogeneous, subadditive functionals-Extension of linear functionals-Gauge functions of convex sets

3.2 Geometric Hahn-Banach theorem,

The hyperplane separation theorem

3.3 Extensions of the Hahn-Banach theorem,

The Agnew-Morse theorem-The

Bohnenblust-Sobczyk-Soukhomlinov theorem

4. Applications of the Hahn-Banach theorem

4.1 Extension of positive linear functionals,

4.2 Banach limits.

4.3 Finitely additive invariant set functions,

Historical note,

5. Normed Linear Spaces

5.1 Norms,

Norms for quotient spaces-Complete normed linear spaces-The spaces C, B-Lp spaces and H61der's inequality-Sobolev spaces, embedding theorems-Separable spaces

5.2 Noncompactness of the unit bail,

Uniform convexity-The Mazur-Ulam theorem on isometrics

5.3 Isometrics,

6. Hilbert Space

6.1 Scalar product,

Schwarz inequality Parallelogram identity——Completeness,closure-e2, L

6.2 Closest point in a closed convex subset, 54Orthogonal complement of a subspace-Orthogonal decomposition

6.3 Linear functionals,

The Riesz-Frechet representation theorem-Lax-Milgram lemma

6.4 Linear span,

Orthogonal projection-Orthonormal bases, Gram-Schmidt process-Isometries of a Hilbert space

7. Applications of Hilbert Space Results

7.1 Radon-Nikodym theorem,

7.2 Dirichlet's problem,

Use of the Riesz-Frechet theorem-Use of the Lax-Milgram theorem Use of orthogonal decomposition

8. Duals of Normed Linear Spaces

8.1 Bounded linear functionals,

Dual space

8.2 Extension of bounded linear functionals,

Dual characterization of norm-Dual characterization of distance from a subspace-Dual characterization of the closed linear span of a set

8.3 Reflexive spaces,

Reflexivity of Lp, 1 < p < -Separable spaces-Separability of the dual-Dual of C(Q), Q compact-Reflexivity of subspaces

8.4 Support function of a set,

Dual characterization of convex hull-Dual characterization of distance from a closed, convex set

9. Applications of Duality

9.1 Completeness of weighted powers,

9.2 The Muntz approximation theorem,

9.3 Runge'stheorem,

9.4 Dual variational problems in function theory,

9.5 Existence of Green's function,

10. Weak Convergence

10.1 Uniform boundedness of weakly convergent sequences, 101 Principle of uniform boundedness-Weakly sequentially closed convex sets

10.2 Weak sequential compactness, 104 Compactness of unit ball in reflexive space

10.3 Weak* convergence, 105 Helly's theorem

11. Applications of Weak Convergence

11.1 Approximation of the function by continuous functions, 108 Toeplitz's theorem on summability

11.2 Divergence of Fourier series,

11.3 Approximate quadrature,

11.4 Weak and strong analyticity of vector-valued functions,

11.5 Existence of solutions of partial differential equations, 112 Galerkin's method

11.6 The representation of analytic functions with positive real part, 115 Hergiotz-Riesz theorem

12. The Weak and Weak* Topologies

Comparison with weak sequential topology-Closed convex sets in the weak topology——Weak compactness-Alaoglu's theorem

13. Locally Convex Topologies and the Krein-Milman Theorem

13.1 Separation of points by linear functionals,

13.2 The Krein-Milman theorem,

13.3 The Stone-Weierstrass theorem,

13.4 Choquet's theorem,

14. Examples of Convex Sets and Their Extreme Points

14.1 Positivefunctionals,

14.2 Convex functions,

14.3 Completely monotone functions,

14.4 Theorems of Caratheodory and Bochner,

14.5 A theorem of Krein,

14.6 Positive harmonic functions,

14.7 The Hamburger moment problem,

14.8 G. Birkhoff's conjecture,

14.9 De Finetti's theorem,

14.10 Measure-preserving mappings,

Historical note,

15. Bounded Linear Maps

15.1 Boundedness and continuity,

Norm of a bounded linear map-Transpose

15.2 Strong and weak topologies,

Strong and weak sequential convergence

15.3 Principle of uniform boundedness,

15.4 Composition of bounded maps,

15.5 The open mapping principle,

Closed graph theorem Historical note,

16. Examples of Bounded Linear Maps

16.1 Boundedness of integral operators,

Integral operators of Hilbert-Schmidt type-Integral operators of Holmgren type

16.2 The convexity theorem of Marcel Riesz,

16.3 Examples of bounded integral operators,

The Fourier transform, Parseval's theorem and Hausdorff-Young inequality-The Hilbert transform The Laplace transform-The Hilbert-Hankel transform

……

A. Riesz-Kakutani representation theorem

B. Theory of distributions

C. Zorn's Lemma

Author Index

Subject Index