希爾伯特空間問(wèn)題集

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.

內(nèi)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

書(shū)籍目錄

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……

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