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
……