不等式

出版者:世界图书出版公司

作者:加林

页数:335

出版时间:2012-1

价格:59.00元

isbn号码:9787510042829

具体描述

《不等式》内容简介：The first intention of this book， then， is to establish fundamental inequalities in this area． But more importantly， its purpose is to put them in context， and to show how useful they are． Although the book is very largely self—contained， it should therefore principally be of interest to analysts， and to those who use analysis seriously。

目录信息

Introduction1 Measure and integral 1.1 Measure 1.2 Measurable functions 1.3 Integration 1.4 Notes and remarks2 The Cauchy-Schwarz inequality 2.1 Cauchy's inequality 2.2 Inner-product spaces 2.3 The Cauchy-Schwarz inequality 2.4 Notes and remarks3 The AM-GM inequality 3.1 The AM-GM inequality 3.2 Applications 3.3 Notes and remarks4 Convexity and Jensen's inequality 4.1 Convex sets and convex functions 4.2 Convex functions on an interval 4.3 Directional derivatives and sublinear functionals 4.4 The Hahn-Banach theorem 4.5 Normed spaces, Banach spaces and Hilbert space 4.6 The Hahn-Banach theorem for normed spaces 4.7 Barycentres and weak integrals 4.8 Notes and remarks5 The Lp spaces 5.1 Lp spaces, and Minkowski's inequality 5.2 The Lebesgue decomposition theorem 5.3 The reverse Minkowski inequality 5.4 HSlder's inequality 5.5 The inequalities of Liapounov and Littlewood 5.6 Duality 5.7 The Loomis-Whitney inequali'ty 5.8 A Sobolev inequality 5.9 Schur's theorem and Schur's test 5.10 Hilbert's absolute inequality 5.11 Notes and remarks6 Banach function spaces 6.1 Banach function spaces 6.2 Function space duality 6.3 Orlicz space 6.4 Notes and remarks7 Rearrangements 7.1 Decreasing rearrangements 7.2 Rearrangement-invariant Banach function spaces 7.3 Muirhead's maximal function 7.4 Majorization 7.5 Calder6n's interpolation theorem and its converse 7.6 Symmetric Banach sequence spaces 7.7 The method of transference 7.8 Finite doubly stochastic matrices 7.9 Schur convexity 7.10 Notes and remarks Maximal inequalities 8.1 The Hardy-Riesz inequality 8.2 The Hardy-Riesz inequality 8.3 Related inequalities 8.4 Strong type and weak type 8.5 Riesz weak type 8.6 Hardy, Littlewood, and a batsman's averages 8.7 Riesz's sunrise lemma 8.8 Differentiation almost everywhere 8.9 Maximal operators in higher dimensions 8.10 The Lebesgue density theorem 8.11 Convolution kernels 8.12 Hedberg's inequality 8.13 Martingales 8.14 Doob's inequality 8.15 The martingale convergence theorem 8.16 Notes and remarks9 Complex interpolation 9.1 Hadamard's three lines inequality 9.2 Compatible couples and intermediate spaces 9.3 The Riesz-Thorin interpolation theorem 9.4 Young's inequality 9.5 The Hausdorff-Young inequality 9.6 Fourier type 9.7 The generalized Clarkson inequalities 9.8 Uniform convexity 9.9 Notes and remarks10 Real interpolation 10.1 The Marcinkiewicz interpolation theorem: I 10.2 Lorentz spaces 10.3 Hardy's inequality 10.4 The scale of Lorentz spaces 10.5 The Marcinkiewicz interpolation theorem: II 10.6 Notes and remarks11 The Hilbert transform, and Hilbert's inequalities 11.1 The conjugate Poisson kernel 11.2 The Hilbert transform on 11.3 The Hilbert transform on 11.4 Hilbert's inequality for sequences 11.5 The Hilbert transform on T 11.6 Multipliers 11.7 Singular integral operators 11.8 Singular integral operators on 11.9 Notes and remarks12 Khintchine's inequality 12.1 The contraction principle 12.2 The reflection principle, and Lavy's inequalities 12.3 Khintchine's inequality 12.4 The law of the iterated logarithm 12.5 Strongly embedded subspaces 12.6 Stable random variables 12.7 Sub-Gaussian random variables 12.8 Kahane's theorem and Kahane's inequality 12.9 Notes and remarks13 Hypercontractive and logarithmic Sobolev inequalities 13.1 Bonami's inequality 13.2 Kahane's inequality revisited 13.3 The theorem of Lataa and Oleszkiewicz 13.4 The logarithmic Sobolev inequality on Dd 13.5 Gaussian measure and the Hermite polynomials 13.6 The central limit theorem 13.7 The Gaussian hypercontractive inequality 13.8 Correlated Gaussian random variables 13.9 The Gaussian logarithmic Sobolev inequality 13.10 The logarithmic Sobolev inequality in higher dimensions 13.11 Beckner's inequality 13.12 The Babenko-Beckner inequality 13.13 Notes and remarks14 Hadamard's inequality 14.1 Hadamard's inequality 14.2 Hadamard numbers 14.3 Error-correcting codes 14.4 Note and remark15 Hilbert space operator inequalities 15.1 Jordan normal form 15.2 Riesz operators 15.3 Related operators 15.4 Compact operators 15.5 Positive compact operators 15.6 Compact operators between Hilbert spaces 15.7 Singular numbers, and the Rayleigh-Ritz minimax formula 15.8 Weyl's inequality and Horn's inequality 15.9 Ky Fan's inequality 15.10 Operator ideals 15.11 The Hilbert-Schmidt class 15.12 The trace class 15.13 Lidskii's trace formula 15.14 Operator ideal duality 15.15 Notes and remarks16 Summing operators 16.1 Unconditional convergence 16.2 Absolutely summing operators 16.3 (p, q)-summing operators 16.4 Examples of p-summing operators 16.5 (p, 2)-summing operators between Hilbert spaces 16.6 Positive operators on 16.7 Mercer's theorem 16.8 p-summing operators between Hilbert spaces 16.9 Pietsch's domination theorem 16.10 Pietsch's factorization theorem 16.11 p-summing operators between Hilbert spaces 16.12 The Dvoretzky-Rogers theorem 16.13 Operators that factor through a Hilbert space 16.14 Notes and remarks17 Approximation numbers and eigenvalues 17.1 The approximation, Gelfand and Weyl numbers 17.2 Subadditive and submultiplicative properties 17.3 Pietsch's inequality 17.4 Eigenvalues of p-summing and (p, 2)-summing endomorphisms 17.5 Notes and remarks18 Grothendieck's inequality, type and cotype 18.1 Littlewood's 4/3 inequality 18.2 Grothendieck's inequality 18.3 Grothendieck's theorem 18.4 Another proof, using Paley's inequality 18.5 The little Grothendieck theorem 18.6 Type and cotype 18.7 Gaussian type and cotype 18.8 Type and cotype of LP spaces 18.9 The little Grothendieck theorem revisited 18.10 More on cotype 18.11 Notes and remarksReferencesIndex of inequalitiesIndex
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用户评价

我是在一个相对放松的暑假里开始阅读《不等式》的，最初只是抱着随便看看的心态，没想到却被它内在的逻辑美感深深吸引住了。这本书最出彩的地方，在我看来，是对“对称性”和“反向思维”在不等式构建中的运用。作者详细阐述了如何利用变量替换和参数变化来简化复杂表达式，从而揭示隐藏在表象之下的简洁关系。比如，书中介绍的那个关于多项式根的Schur不等式，一开始看起来极其复杂，但通过作者引导的变量代换，瞬间变得清晰可辨。这种从繁复走向简约的过程，如同剥洋葱一样，每剥开一层，都能看到更纯粹的核心。此外，这本书的语言风格非常具有感染力，它不使用那些生硬的数学术语来刻意制造距离感，而是用一种仿佛在与同好探讨的语气，邀请读者共同进入这个充满约束与平衡的数学领域。对于那些对“美”有追求的数学学习者来说，《不等式》提供了一种结构之美、逻辑之美和证明之美。

这本书的阅读体验，坦率地说，是充满挫败感，但也伴随着巨大的喜悦。它绝不是那种可以轻松翻阅的休闲读物。许多证明步骤，尤其是涉及到一些高阶不等式，需要我反复推敲，甚至需要借助其他参考书来辅助理解作者跳跃的思维。但这恰恰是它的魅力所在。作者似乎不满足于提供最平庸的证明路径，他总是在探寻那些最精妙、最简洁的论证。这种对“优雅”的追求贯穿始终。我尤其欣赏书中对“反证法”在处理某些极端不等式时的应用分析，作者通过构建反例，清晰地界定了特定不等式的适用边界，这对于避免在实际应用中误用公式至关重要。总而言之，这本书是为那些不惧怕挑战、渴望挑战自身智力极限的读者准备的。它要求读者付出时间、专注和毅力，但作为回报，它会赠予你数学世界中最深刻的洞察力之一。

老实说，我买这本书的初衷是想找一本能帮我快速提高竞赛水平的工具书，但《不等式》带给我的远不止是“技巧”。它更像是一部数学思想的史诗。它的叙事方式非常独特，不是那种冷冰冰的教科书风格，反而有一种历史的厚重感和探索的激情。作者似乎在引导我们重走数学家们发现和证明这些不等式的艰辛历程。书中对一些著名不等式的“诞生记”描写得绘声绘色，比如伽马函数的引入如何简化了某些复杂积分不等式的证明，这让我对数学的发展脉络有了更深的理解。我印象特别深的是关于函数凹凸性与不等式之间联系的章节，作者用非常细腻的笔触描绘了“Jensen不等式”的威力，这种将微积分工具与代数不等式完美融合的方法，展现了数学学科内部的和谐统一性。对于想要深入理解数学本质，而非仅仅停留在解题表面的读者，这本书提供的深度和广度是无与伦比的。它挑战了我的思维定势，迫使我用更灵活、更具创造性的方式去思考问题。

这本《不等式》简直是数学爱好者的福音，我一口气读完了，那种感觉就像是找到了一把能解锁高等数学大门的钥匙。作者在讲解基础概念时，那种深入浅出的功力实在令人佩服。他没有仅仅停留在公式的罗列上，而是花费了大量篇幅去阐述每一个不等式符号背后蕴含的几何意义和逻辑推导，这一点对我这种偏爱直观理解的学习者来说太重要了。比如，在谈到柯西-施瓦茨不等式时，我仿佛能看到两个向量在多维空间中相互“制约”的画面，而不是仅仅记住了那个开根号的复杂表达式。书中对各种常见不等式的证明方法进行了详尽的梳理，从初等的代数技巧到更抽象的分析工具，层次分明，循序渐进。我特别喜欢其中关于“极值”的讨论，作者巧妙地将代数不等式与实际问题（比如资源分配、效率优化）结合起来，让人深刻体会到数学的实用价值。读完后，我感觉自己看待很多日常现象都有了一种新的、更加精确的“数学视角”，不再满足于模糊的描述，而是追求更严格的量化关系。这本书绝对值得放在书架上，时不时翻阅，每次都会有新的感悟。

这本书的排版和装帧虽然朴实，但内容质量却是顶级的。我是一名从事工程设计工作的工程师，工作中经常需要对误差范围和性能上限进行评估，过去我主要依赖经验和一些固定的公式手册。然而，《不等式》让我找到了一个更系统、更可靠的理论基础。书中对“三角不等式”的推广和应用分析得极为透彻，特别是它如何被应用到误差分析和信号处理的边界设定中，那些例子非常贴合实际工作场景。我发现，许多过去感觉“理所当然”的工程假设，在书中都有了严密的数学论证支持。它不仅仅是教你如何证明，更重要的是教你如何“选择”正确的工具。作者在每章末尾设置的“拓展思考题”极富启发性，它们不是那种标准化的、有固定答案的习题，而是引导读者去探索特定不等式在不同数学分支（如概率论、泛函分析）中的变体和应用潜力。这本书的价值在于，它能将抽象的数学概念转化为强大的、可操作的分析工具箱。