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. 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 condensed 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.

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 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 Hiibert 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 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 completeness

4. 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 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. 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 transformations

7.MULTIPLICATION OPERATORS

 61. Diagonal operators

 62. Multiplications on 12

 63. Spectrum of a diagonal operator

 64. Norm of a multiplication

 ……