希尔伯特空间问题集

出版时间：2009-4  出版社：世界图书出版公司  作者：哈尔莫斯  页数：369  
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前言

　　The only way to learn mathematics iS to do mathematics.That tenet iS the foundation of the do.it.yourself，Socratic，or Texas method。the method in which the teacher plays the role of an omniscient but largely uncommuni. cative referoe between the learner and the facts.Although that method iS usually and perhaps necessarily oral。this book tries to use the same method to give a written exposition of certain topics in Hilben space theory.　　The right way to read mathematics iS first to read the definitions of the concepts and the statements of the theorems，and then，putting the book aside，to try to discover the appropriate proofs.If the theorems are not trivial，the attempt might fail，but it iS likely to bc instructive just the same. To the passive reader a routine computation and a miracle of ingenuity come with equal ease，and later，when he must depend on himself。he will find that they went as easily as they came.The active reader，who has found out what does not work.iS in a much better position to understand the reason ror the SUCCESS of the author’S method，and。Iater，to find answers that are not in books.　　This book was written for the active reader.Thc first part consists of problems，frequently preceded by definitions and motivation。and some. times followed by corollaries and historical remarks.Most of the problems are statements to be proved，but some are questions（is it？。what is？），and some are challenges（construct，determine）.The second part，a very short one，consists of hints.A hint iS a word。or a paragraph，usually intended to help the reader find a solution.The hint itself iS not necessarily a con. densed solution of the problem：it may iust point to what I regard as the heart of the matter.Sometimes a problem contains a trap，and the hint may serve to chide the reader for rushing in too recklessly.The third part.

内容概要

　　This book was written for the active reader. The first part consists of problems， frequently preceded by definitions and motivation， and some-times followed by corollaries and historical remarks. Most of the problems are statements to be proved， but some are questions (is it?， what is?)， and some are challenges (construct， determine). The second part， a very short one， consists of hints. A hint is a word， or a paragraph， usually intended to help the reader find a solution. The hint itself is not necessarily a con-densed solution of the problem; it may just point to what I regard as the heart of the matter. Sometimes a problem contains a trap， and the hint may serve to chide the reader for rushing in too recklessly. The third part， the longest， consists of solutions： proofs， answers， or constructions， depending on the nature of the problem

书籍目录

1. VECTORS  1. Limits of quadratic forms  2. Schwarz inequality  3. Representation of linear functionals  4. Strict convexity  5. Continuous curves  6. Uniqueness of crinkled arcs  7. Linear dimension  8. Total sets  9. Infinitely total sets　10. Infinite Vandermondes　11. T-total sets　12. Approximate bases2. SPACES　13. Vector sums　14. Lattice of subspaces　15. Vector sums and the modular law　16. Local compactness and dimension　17. Separability and dimension　18. Measure in Hiibert space3.  WEAK TOPOLOGY　19. Weak closure of subspaces　20. Weak continuity of norm and inner product　21. Semicontinuity of norm　22. Weak separability　23. Weak compactness of the unit bali　24. Weak metrizabilitv of the unit ball　25. Weak closure of the unit sphere　26. Weak metrizability and separability　27. Uniform boundedness　28. Weak metrizability of Hilbert space　29. Linear functionals on /2　30. Weak completeness4.　ANALYTIC  FUNCTIONS　31. Analytic Hilbert spaces　32. Basis for Ae　33. Real functions in He　34. Products in H2　35. Analytic characterization of H2　36. Functional Hilbcrt spaces　37. Kernel functions　38. Conjugation in functional Hilbert spaces　39. Continuity of extension　40. Radial limits　41. Bounded approximation　42. Multiplicativity of extension　43. Dirichlet problem5.  INFINITE MATRICES　44. Column-finite matrices　45. Schur test　46. Hilbert matrix　47. Exponential Hilbert matrix　48. Positivity of the Hilbert matrix　49. Series of vectors6. BOUNDEDNESS AND  INVERTIBILITY　50. Boundedness on bases　51. Uniform boundedness of linear transformations　52. lnvertible transformations　53. Diminishable complements　54. Dimension in inner-product spaces　55. Total orthonormal sets　56. Preservation of dimension　57. Projections of equal rank　58. Closed graph theorem　59. Range inclusion and factorization　60. Unbounded symmetric transformations7.MULTIPLICATION  OPERATORS　61. Diagonal operators　62. Multiplications on 12　63. Spectrum of a diagonal operator　64. Norm of a multiplication……

用户评论 (总计3条)

•   这本书是Halmos教授的名著，他还有好几本名著。本书英文原版是著名的GTM丛书第19本，适合数学研究生阅读。对研究希尔伯特空间和泛函分析的读者，本书很有价值。
•   玻姆哥本哈根解释提出了反建议，德布罗意也在某种程度上采纳了这种见解。玻姆把粒子看作是“客观实在的”结构，就象牛顿力学中的质点一样。位形空间中的波在他的解释中也是“客观实在的”，就象电场一样。位形空间是牵涉到属于系统的全部粒子的不同坐标的一个多维空间。遇到了第一个困难：说位形空间中的波是“实在的”，究竟是什么意协这个空间是一个很抽象的空间。“实在的”一词起源于拉丁字“res”（实体），它的意思是“物”；但物是存在于通常的三维空间中，而不是存在于抽象的位形空间中的。当人们想说位形空间中的波与任何观测者无关时，人们可以说这些波是“客观的”；但人们很难说它们是“实在的”，除非人们甘愿改变这个词的含义。玻姆进一步规定恒波相面的法线是粒子的可能轨道。按照他的想法，这些法线中哪一条是“实在的”轨道取决于系统和测量仪器的历史，并且如果对系统与测量仪器的了解不比实际上能了解的更多的话，“实在的”轨道就无法确定。这种历史实际上包含了隐参量，它就是实验开始以前的“实际”轨道。
  
  泡利（Pauli）所强调指出的，这种解释的一个结果是：许多原子中的一些基态电子应当是静止的，不环绕原子核作任何轨道运动。这似乎和实验相矛盾，因为对基态中电子速度的测量（例如，用康普顿效应的方法），总是显示出基态中有一个速度分布，它由动量空间或速度空间中的波国数的平方所给出——这符合于量子力学定则。但是，这里玻姆能够辩解说，这时测量已经不能再用普通定律来估算了。他同意测量的正常估算确实会得出速度分布；但当考虑到关于测量仪器的量子论——特别是由玻姆在这方面引入的某些奇特的量子势时，那么，电子老是“实在地”静止着的陈述是讲得通的。在粒子位置的测量中，玻姆认为实验的通常解释是正确的；而在速度测量中，他拒绝了通常的解释。以此为代价，玻姆认为他自己有权利主张：“我们不必在量子论的领域中放弃单个系统的准确、合理和客观的描述。”然而，这种客观描述本身却象是一种“意识形态的上层建筑”，它与直接的物理实在关系很少；因为如果量子论保持不变的话，玻姆解释中的隐参量就是永远不能在实在过程的描述中出现的那样一种东西。
  
  为了避免这种困难，玻姆实际上表达了这样一个希望：将来在基本粒子的领域的实验中，隐参量可能会起一部分物理作用，而量子论将因此被证明为错误的。在讲到这样一些奇怪的希望时，玻尔常常说它们在结构上就象是这样的一些句子：“我们可以希望以后会证明有时2X2=5，因为这对我们的财务大有好处。”实际上玻姆希望的满足，将不仅从下面挖掉量子论的基础，并且也挖掉了玻姆解释的基础。当然，同时也必须强调指出，刚才所说的类比，虽然十分恰当，但并不表示将来象玻姆所建议的那样来改变量子论的论证，在逻辑上也是行不通的。因为这不是根本不可想象的，譬如说，未来数理逻辑的扩展，可能给在特殊情况下2X2=5这样的陈述以某种意义，并且这种扩展了的数学甚至可能在经济领域的计算中得到应用。然而，即使提不出令人信服的逻辑根据，我们实际上仍相信，数学中这样的变化在财务上对我们也毫无帮助。因此，很难理解，玻姆的著作所指出的那些可能实现他的希望的数学倡议如何能够用来描述物理现象。
•   呵呵，相当好的书，可惜没有来的急看了