泛函分析(影印版)

出版时间:2007-2  出版社:高等教育出版社  作者:Peter D. Lax  页数:580  
Tag标签:无  

内容概要

  Banach空间、Hilbert空间和线性拓扑空间的基本概念和性质,线性拓扑空间中端点集的性质,有界线性算子的性质等,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Banach代数的Gelfand理论,不变子空间的理论等。

作者简介

作者:(美国)拉克斯

书籍目录

Foreword1. Linear SpacesAxioms for linear spaces-Infinite-dimensional examples-Subspace, linear span-Quotient space-Isomorphism-Convex sets-Extreme subsets2. Linear Maps2.1 Algebra of linear maps,Axioms for linear maps-Sums and composites-Invertible linear maps-Nullspace and range-Invariant subspaces2.2. Index of a linear map,Degenerate maps-Pseudoinverse-IndexmProduct formula for the index-Stability of the index3. The Hahn,Banach Theorem3.1 The extension theorem,Positive homogeneous, subadditive functionals-Extension of linear functionals-Gauge functions of convex sets3.2 Geometric Hahn-Banach theorem,The hyperplane separation theorem3.3 Extensions of the Hahn-Banach theorem,The Agnew-Morse theorem-TheBohnenblust-Sobczyk-Soukhomlinov theorem4. Applications of the Hahn-Banach theorem4.1 Extension of positive linear functionals,4.2 Banach limits.4.3 Finitely additive invariant set functions,Historical note,5. Normed Linear Spaces5.1 Norms,Norms for quotient spaces-Complete normed linear spaces-The spaces C, B-Lp spaces and H61der's inequality-Sobolev spaces, embedding theorems-Separable spaces5.2 Noncompactness of the unit bail,Uniform convexity-The Mazur-Ulam theorem on isometrics5.3 Isometrics,6. Hilbert Space6.1 Scalar product,Schwarz inequality Parallelogram identity——Completeness,closure-e2, L6.2 Closest point in a closed convex subset, 54Orthogonal complement of a subspace-Orthogonal decomposition6.3 Linear functionals,The Riesz-Frechet representation theorem-Lax-Milgram lemma6.4 Linear span,Orthogonal projection-Orthonormal bases, Gram-Schmidt process-Isometries of a Hilbert space7. Applications of Hilbert Space Results7.1 Radon-Nikodym theorem,7.2 Dirichlet's problem,Use of the Riesz-Frechet theorem-Use of the Lax-Milgram theorem Use of orthogonal decomposition8. Duals of Normed Linear Spaces8.1 Bounded linear functionals,Dual space8.2 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 set8.3 Reflexive spaces,Reflexivity of Lp, 1 < p < -Separable spaces-Separability of the dual-Dual of C(Q), Q compact-Reflexivity of subspaces8.4 Support function of a set,Dual characterization of convex hull-Dual characterization of distance from a closed, convex set9. Applications of Duality9.1 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 Convergence10.1 Uniform boundedness of weakly convergent sequences, 101 Principle of uniform boundedness-Weakly sequentially closed convex sets10.2 Weak sequential compactness, 104 Compactness of unit ball in reflexive space10.3 Weak* convergence, 105 Helly's theorem11. Applications of Weak Convergence11.1 Approximation of the function by continuous functions, 108 Toeplitz's theorem on summability11.2 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 method11.6 The representation of analytic functions with positive real part, 115 Hergiotz-Riesz theorem12. The Weak and Weak* TopologiesComparison with weak sequential topology-Closed convex sets in the weak topology——Weak compactness-Alaoglu's theorem13. Locally Convex Topologies and the Krein-Milman Theorem13.1 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 Points14.1 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 Maps15.1 Boundedness and continuity,Norm of a bounded linear map-Transpose15.2 Strong and weak topologies,Strong and weak sequential convergence15.3 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 Maps16.1 Boundedness of integral operators,Integral operators of Hilbert-Schmidt type-Integral operators of Holmgren type16.2 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 theoremB. Theory of distributionsC. Zorn's LemmaAuthor IndexSubject Index

章节摘录

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用户评论 (总计14条)

 
 

  •     丰富的内容,完美的解说,精彩的论证,流畅的表述,必将使这本诞生不久的泛函分析成为一部不可多得的优秀杰作。后一章基本是前一章的应用延伸和拓广,抽象的泛函分析理论有最直接的数学背景。
  •     影印版,但是学生么,用着还不错的说,很实惠。
  •     感觉都是一笔带过的样子,就是讲得内容挺多挺前沿的,打基础的话还不如柯尔莫哥洛夫那本《函数论与泛函分析初步》
  •     内容好,印刷清晰。爱不释手。
  •     本书作者是Wolf数学奖得主,这本书写得也很认真,内容也很丰富,但也比较难些。不过到现在我只是看了很少一部分。以后再继续看吧。
  •     包装像是掉到泥里边了,书的下角卷起来了,强烈抗议
  •     都说好,那就买来看一下!院士的书还是值得读的!
  •     卖来镇宅的,内容还没看,物流不给力啊!
  •     一句话:性价比很高。
  •     书很经典,认真研读之后视野放得更宽
  •     美国科学院院士的著作,写的简明,值得收藏
  •     听别人推荐过此书非常值得研究,就买了一本。。。很好
  •     确实很好。这些书如果看翻译的,可能还没有原本容易理解。
  •        这本泛函分析要求的基础比较高,是一本锻炼研究能力的研究生教材,一些定理的证明比较简略,需要留心。习题比较少,有点像一张大地图,但需要读者丰富。
 

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