Book:George F. Simmons/Introduction to Topology and Modern Analysis

Subject Matter

 * Topology
 * Analysis

Contents

 * Preface
 * A Note to the Reader


 * PART ONE: TOPOLOGY


 * Chapter One $\quad$ SETS AND FUNCTIONS
 * 1. Sets and set inclusion
 * 2. The algebra of sets
 * 3. Functions
 * 4. Products of sets
 * 5. Partitions and equivalence relations
 * 6. Countable sets
 * 7. Uncountable sets
 * 8. Partially ordered sets and lattices


 * Chapter Two $\quad$ METRIC SPACES
 * 9. The definition and some examples
 * 10. Open sets
 * 11. Closed sets
 * 12. Convergence, completeness, and Baire's theorem
 * 13. Continuous mappings
 * 14. Spaces of continuous functions
 * 15. Euclidean and unitary spaces


 * Chapter Three $\quad$ TOPOLOGICAL SPACES
 * 16. The definition and some examples
 * 17. Elementary concepts
 * 18. Open bases and open subbases
 * 19. Weak topologies
 * 20. The function algebras $\mathscr C \left({X, R}\right)$ and $\mathscr C \left({X, C}\right)$


 * Chapter Four $\quad$ COMPACTNESS
 * 21. Compact spaces
 * 22. Products of spaces
 * 23. Tychonoff's theorem and locally compact spaces
 * 24. Compactness for metric spaces
 * 25. Ascoli's theorem


 * Chapter Five $\quad$ SEPARATION
 * 26. $T_1$-spaces and Hausdorff spaces
 * 27. Completely regular spaces and normal spaces
 * 28. Urysohn's lemma and the Tietze expansion theorem
 * 29. The Urysohn imbedding theorem
 * 30. The Stone-Čech compactification


 * Chapter Six $\quad$ CONNECTEDNESS
 * 31. Connected spaces
 * 32. The components of a space
 * 33. Totally disconnected spaces
 * 34. Locally connected spaces


 * Chapter Seven $\quad$ APPROXIMATION
 * 35. The Weierstrass approximation theorem
 * 36. The Stone-Weierstrass theorem
 * 37. Locally compact Hausdorff spaces
 * 38. The extended Stone-Weierstrass theorem


 * PART TWO: OPERATORS


 * Chapter Eight $\quad$ ALGEBRAIC SYSTEMS
 * 39. Groups
 * 40. Rings
 * 41. The structure of rings
 * 42. Linear spaces
 * 43. The dimension of a linear space
 * 44. Linear transformations
 * 45. Algebras


 * Chapter Nine $\quad$ BANACH SPACES
 * 46. The definition and some examples
 * 47. Continuous linear transformations
 * 48. The Hahn-Banach theorem
 * 49. The natural embedding of $N$ in $N^{**}$
 * 50. The open mapping theorem
 * 51. The conjugate of an operator


 * Chapter Ten $\quad$ HILBERT SPACES
 * 52. The definition and some simple properties
 * 53. Orthogonal complements
 * 54. Orthonormal sets
 * 55. The conjugate space $H^*$
 * 56. The adjoint of an operator
 * 57. Self-adjoint operators
 * 58. Normal and unitary operators
 * 59. Projections


 * Chapter Eleven $\quad$ FINITE-DIMENSIONAL SPECTRAL THEORY
 * 60. Matrices
 * 61. Determinants and the spectrum of an operator
 * 62. The spectral theorem
 * 63. A survey of the situation


 * PART THREE: ALGEBRAS OF OPERATORS


 * Chapter Twelve $\quad$ GENERAL PRELIMINARIES ON BANACH ALGEBRAS
 * 64. The definition and some examples
 * 65. Regular and singular elements
 * 66. Topological divisors of zero
 * 67. The spectrum
 * 68. The formula for the spectral radius
 * 69. The radical and semi-simplicity


 * Chapter Thirteen $\quad$ THE STRUCTURE OF COMMUTATIVE BANACH ALGEBRAS
 * 70. The Gelfand mapping
 * 71. Applications of the formula $r \left({x}\right) = \lim \left\Vert{x^n}\right\Vert^{1/n}$
 * 72. Involutions in Banach algebras
 * 73. The Gelfand-Neumark theorem


 * Chapter Fourteen $\quad$ SOME SPECIAL COMMUTATIVE BANACH ALGEBRAS
 * 74. Ideals in $\mathscr C \left({X}\right)$ and the Banach-Stone theorem
 * 75. The Stone-Čech compactification (continued)
 * 76. Commitative $C^*$-algebras


 * APPENDICES


 * ONE $\quad$ Fixed point theorems and some applications to analysis
 * TWO $\quad$ Continuous curves and the Hahn-Mazurkiewicz theorem
 * THREE $\quad$ Boolean algebras, Boolean rings, and Stone's theorem


 * Bibliography


 * Index of Symbols


 * Subject Index