A Hilbert Space Problem Book

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