基本信息

書名:天元基金數學叢書：泛函分析

定價：46.40元

作者:[美] 拉剋斯 著

齣版社：高等教育齣版社

齣版日期：2007-02-01

ISBN：9787040216493

字數：

頁碼：580

版次：1

裝幀：平裝

開本：16開

商品重量：

編輯推薦

暫無相關內容

目錄

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 extensiotheorem,
Positive homogeneous, subadditive functionals-Extensioof linear functionals-Gauge functions of convex sets
3.2 Geometric Hahn-Banach theorem,
The hyperplane separatiotheorem
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 Extensioof 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 H61ders inequality-Sobolev spaces, embedding theorems-Separable spaces
5.2 Noncompactness of the unit bail,
Uniform convexity-The Mazur-Ulam theorem oisometrics
5.3 Isometrics,

6. Hilbert Space
6.1 Scalar product,
Schwarz inequality Parallelogram identity——Completeness,closure-e2, L
6.2 Closest point ia closed convex subset, 54Orthogonal complement of a subspace-Orthogonal decomposition
6.3 Linear functionals,
The Riesz-Frechet representatiotheorem-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 Dirichlets 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 Extensioof bounded linear functionals,
Dual characterizatioof norm-Dual characterizatioof distance from a subspace-Dual characterizatioof the closed linear spaof 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 functioof a set,
Dual characterizatioof convex hull-Dual characterizatioof distance from a closed, convex set

9. Applications of Duality
9.1 Completeness of weighted powers,
9.2 The Muntz approximatiotheorem,
9.3 Rungestheorem,
9.4 Dual variational problems ifunctiotheory,
9.5 Existence of Greens 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 ireflexive space
10.3 Weak* convergence, 105 Hellys theorem

11. Applications of Weak Convergence
11.1 Approximatioof the functioby continuous functions, 108 Toeplitzs theorem osummability
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 Galerkins method
11.6 The representatioof analytic functions with positive real part, 115 Hergiotz-Riesz theorem

12. The Weak and Weak* Topologies
Comparisowith weak sequential topology-Closed convex sets ithe weak topology——Weak compactness-Alaoglus theorem

13. Locally Convex Topologies and the Krein-MilmaTheorem
13.1 Separatioof points by linear functionals,
13.2 The Krein-Milmatheorem,
13.3 The Stone-Weierstrass theorem,
13.4 Choquets 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. Birkhoffs conjecture,
14.9 De Finettis 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 Compositioof bounded maps,
15.5 The opemapping 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 Holmgretype
16.2 The convexity theorem of Marcel Riesz,
16.3 Examples of bounded integral operators,
The Fourier transform, Parsevals theorem and Hausdorff-Young inequality-The Hilbert transform The Laplace transform-The Hilbert-Hankel transform
……
A. Riesz-Kakutani representatiotheorem
B. Theory of distributions
C. Zorns Lemma
Author Index
Subject Index

內容提要

《泛函分析（版）》是美國科學院院士Peter D．Lax在CotJrant數學所長期講授泛函分析課程的教學經驗基礎上編寫的。《泛函分析（版）》括泛函分析的基本內容：Barlach空間、Hilbert空間和綫性拓撲空間的基本概念和性質，綫性拓撲空間中的凸集及其端點集的性質，有界綫性算子的性質等。可作為本科生泛函分析課的教學內容；還括泛函分析較深的內容：自伴算子的譜分解理論。緊算子的理論，交換Barlach代數的Gelfand理論，不變子空間的理論等。可作為研究生泛函分析課的教學內容。《泛函分析（版）》特彆強調泛函分析與其他數學分支的聯係及泛函分析理論的應用，可以使讀者深刻地理解到：抽象的泛函分析理論有著豐富的數學背景。

文摘

暫無相關內容

作者介紹

暫無相關內容