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A Hilbert Space Problem Book
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A Hilbert Space Problem Book

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P. R. Halmos
754 g
235x155x26 mm

Written for the active reader with some background in the topic, this book presents problems in Hilbert space theory, with definitions, corollaries and historical remarks, hints, proofs, answers and constructions.
1. Vectors.- 1. Limits of quadratic forms.- 2. Schwarz inequality.- 3. Representation of linear functional.- 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-totalsets.- 12. Approximate bases.- 2. 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 Hilbert space.- 3. 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 ball.- 24. Weak metrizability 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 l2.- 30. Weak completeness.- 4. Analytic Functions.- 31. Analytic Hilbert spaces.- 32. Basis for A2.- 33. Real functions in H2.- 34. Products in H2.- 35. Analytic characterization of H2.- 36. Functional Hilbert 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 problem.- 5. 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 vectors.- 6. Boundedness and Invertibility.- 50. Boundedness on bases.- 51. Uniform boundedness of linear transformations.- 52. Invertible transformations.- 53. Diminishablc 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 transformations.- 7. Multiplication Operators.- 61. Diagonal operators.- 62. Multiplications on l2.- 63. Spectrum of a diagonal operator.- 64. Norm of a multiplication.- 65. Boundedness of multipliers.- 66. Boundedness of multiplications.- 67. Spectrum of a multiplication.- 68. Multiplications on functional Hilbert spaces.- 69. Multipliers of functional Hilbert spaces.- 8. Operator Matrices.- 70. Commutative operator determinants.- 71. Operator determinants.- 72. Operator determinants with a finite entry.- 9. Properties of Spectra.- 73. Spectra and conjugation.- 74. Spectral mapping theorem.- 75. Similarity and spectrum.- 76. Spectrum of a product.- 77. Closure of approximate point spectrum.- 78. Boundary of spectrum.- 10. Examples of Spectra.- 79. Residual spectrum of a normal operator.- 80. Spectral parts of a diagonal operator.- 81. Spectral parts of a multiplication.- 82. Unilateral shift.- 83. Structure of the set of eigenvectors.- 84. Bilateral shift.- 85. Spectrum of a functional multiplication.- 11. Spectral Radius.- 86. Analyticity of resolvents.- 87. Non-emptiness of spectra.- 88. Spectral radius.- 89. Weighted shifts.- 90. Similarity of weighted shifts.- 91. Norm and spectral radius of a weighted shift.- 92. Power norms.- 93. Eigenvalues of weighted shifts.- 94. Approximate point spectrum of a weighted shift.- 95. Weighted sequence spaces.- 96. One-point spectrum.- 97. Analytic quasinilpotents.- 98. Spectrum of a direct sum.- 12. Norm Topology.- 99. Metric space of operators.- 100. Continuity of inversion.- 101. Interior of conjugate class.- 102. Continuity of spectrum.- 103. Semicontinuity of spectrum.- 104. Continuity of spectral radius.- 105. Normal continuity of spectrum.- 106. Quasinilpotent perturbations of spectra.- 13. Operator Topologies.- 107. Topologies for operators.- 108. Continuity of norm.- 109. Semicontinuity of operator norm.- 110. Continuity of adjoint.- 111. Continuity of multiplication.- 112. Separate continuity of multiplication.- 113. Sequential continuity of multiplication.- 114. Weak
From the Preface: "This book was written for the active reader. The first part consists of problems, frequently preceded by definitions and motivation, and sometimes followed by corollaries and historical remarks... The second part, a very short one, consists of hints... The third part, the longest, consists of solutions: proofs, answers, or contructions, depending on the nature of the problem....This is not an introduction to Hilbert space theory. Some knowledge of that subject is a prerequisite: at the very least, a study of the elements of Hilbert space theory should proceed concurrently with the reading of this book."

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