泛函分析 扫描版[DJVU]

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中文名: 泛函分析

原名: Functional Analysis

作者: Lax

图书分类: 科技

资源格式: DJVU

版本: 扫描版

出版社: Wiley

书号: 0-471-55604-1

发行时间: 2002年

地区: 美国

语言: 英文

简介:

 

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, 8

axioms for linear maps-sums and composites-invertible linear maps-nullspace and range-invariant subspaces

2.2. index of a linear map, 12

degenerate maps-pseudoinverse-indexmproduct formula for the index-stability of the index

3. the hahn,banach theorem

3.1 the extension theorem, 19

positive homogeneous, subadditive functionals-extension of linear functionals-gauge functions of convex sets

3.2 geometric hahn-banach theorem, 21

the hyperplane separation theorem

3.3 extensions of the hahn-banach theorem, 24

the agnew-morse theorem-the

bohnenblust-sobczyk-soukhomlinov theorem

4. applications of the hahn-banach theorem

.4.1 extension of positive linear functionals, 29

4.2 banach limits. 31

4.3 finitely additive invariant set functions, 33

historical note, 34

5. normed linear spaces

5.1 norms, 36

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, 43

uniform convexity-the mazur-ulam theorem on isometrics

5.3 isometrics, 47

6. hilbert space

6.1 scalar product, 52

schwarz inequality parallelogram identity——completeness,closure-e2, l2

6.2 closest point in a closed convex subset, 54orthogonal complement of a subspace-orthogonal decomposition

6.3 linear functionals, 56

the riesz-frechet representation theorem-lax-milgram lemma

6.4 linear span, 58

orthogonal projection-orthonormal bases, gram-schmidt process-isometries of a hilbert space

7. applications of hilbert space results

7.1 radon-nikodym theorem, 63

7.2 dirichlet’s problem, 65

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, 72

dual space

8.2 extension of bounded linear functionals, 74

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, 78

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, 83

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

9. applications of duality

9.1 completeness of weighted powers, 87

9.2 the muntz approximation theorem, 88

9.3 runge’stheorem, 91

9.4 dual variational problems in function theory, 91

9.5 existence of green’s function, 94

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, 109

11.3 approximate quadrature, 110

11.4 weak and strong analyticity of vector-valued functions, 111

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, 123

13.2 the krein-milman theorem, 124

13.3 the stone-weierstrass theorem, 126

13.4 choquet’s theorem, 128

14. examples of convex sets and their extreme points

14.1 positivefunctionals, 133

14.2 convex functions, 135

14.3 completely monotone functions, 137

14.4 theorems of caratheodory and bochner, 141

14.5 a theorem of krein, 147

14.6 positive harmonic functions, 148

14.7 the hamburger moment problem, 150

14.8 g. birkhoff’s conjecture, 151

14.9 de finetti’s theorem, 156

14.10 measure-preserving mappings, 157

historical note, 159

15. bounded linear maps

15.1 boundedness and continuity, 160

norm of a bounded linear map-transpose

15.2 strong and weak topologies, 165

strong and weak sequential convergence

15.3 principle of uniform boundedness, 166

15.4 composition of bounded maps, 167

15.5 the open mapping principle, 168

closed graph theorem historical note, 172

16. examples of bounded linear maps

16.1 boundedness of integral operators, 173

integral operators of hilbert-schmidt type-integral operators of holmgren type

16.2 the convexity theorem of marcel riesz, 177

16.3 examples of bounded integral operators, 180

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

内容简介：

《泛函分析》是美国科学院院士Peter D．Lax在 Courant 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》包括泛函分析的基本内容：Banach 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》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。

内容截图：

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