Chapter 1. Introduction Chapter 2. Categories Chapter 3. The Category of Groups Chapter 4. Subgroups Chapter 5. Normal Subgroups Chapter 6. Homomorphisms Chapter 7. Direct Products and Sums of Groups Chapter 8. Relations Chapter 9. The Category of Vector Spaces Chapter 10. Subspaces Chapter 11. Linear Mappings; Direct Products and Sums Chapter 12. From Real to Complex Vector Spaces and Back Chapter 13. Duals Chapter 14. Multilinear Mappings; Tensor Products Chapter 15. Example: Minkowski Vector Space Chapter 16. Example: The Lorentz Group Chapter 17. Functors Chapter 18. The Category of Associative Algebras Chapter 19. The Category of Lie Algebras Chapter 20. Example: The Algebra of Observables Chapter 21. Example: Fock Vector Space Chapter 22. Representations: General Theory Chapter 23. Representations on Vector Spaces Chapter 24. The Algebraic Categories: Summary Chapter 25. Subsets and Mappings Chapter 26. Topological Spaces Chapter 27. Continuous Mappings Chapter 28. The Category of Topological Spaces Chapter 29. Nets Chapter 30. Compactness Chapter 31. The Compact-Open Topology Chapter 32. Connectedness Chapter 33. Example: Dynamical Systems Chapter 34. Homotopy Chapter 35. Homology Chapter 36. Homology: Relation to Homotopy Chapter 37. The Homology Functors Chapter 38. Uniform Spaces Chapter 39. The Completion of a Uniform Space Chapter 40. Topological Groups Chapter 41. Topological Vector Spaces Chapter 42. Categories: Summary Chapter 43. Measure Spaces Chapter 44. Constructing Measure Spaces Chapter 45. Measurable Functions Chapter 46. Integrals Chapter 47. Distributions Chapter 48. Hilbert Spaces Chapter 49. Bounded Operators Chapter 50. The Spectrum of a Bounded Operator Chapter 51. The Spectral Theorem: Finite-dimensional Case Chapter 52. Continuous Functions of a Hermitian Operator Chapter 53. Other Functions of a Hermitian Operator Chapter 54. The Spectral Theorem Chapter 55. Operators (Not Necessarily Bounded) Chapter 56. Self-Adjoint Operators