Book:George F. Simmons/Introduction to Topology and Modern Analysis
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George F. Simmons: Introduction to Topology and Modern Analysis
Published $\text {1963}$, McGraw-Hill
Subject Matter
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 One $\quad$ SETS AND FUNCTIONS
- 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 Two $\quad$ METRIC 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 Three $\quad$ TOPOLOGICAL SPACES
- 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 Four $\quad$ COMPACTNESS
- 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 Five $\quad$ SEPARATION
- Chapter Six $\quad$ CONNECTEDNESS
- 31. Connected spaces
- 32. The components of a space
- 33. Totally disconnected spaces
- 34. Locally connected spaces
- Chapter Six $\quad$ CONNECTEDNESS
- 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
- Chapter Seven $\quad$ APPROXIMATION
- 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 Eight $\quad$ ALGEBRAIC SYSTEMS
- 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 Nine $\quad$ BANACH SPACES
- 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 Ten $\quad$ HILBERT SPACES
- 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
- Chapter Eleven $\quad$ FINITE-DIMENSIONAL SPECTRAL THEORY
- 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 Twelve $\quad$ GENERAL PRELIMINARIES ON BANACH ALGEBRAS
- 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 Thirteen $\quad$ THE STRUCTURE OF COMMUTATIVE BANACH ALGEBRAS
- 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
- Chapter Fourteen $\quad$ SOME SPECIAL COMMUTATIVE BANACH 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
Cited by
- 1975: Bert Mendelson: Introduction to Topology (3rd ed.)
- 1975: W.A. Sutherland: Introduction to Metric and Topological Spaces
- 1977: K.G. Binmore: Mathematical Analysis: A Straightforward Approach
Sources
- 1975: Bert Mendelson: Introduction to Topology (3rd ed.) ... (previous) ... (next): Chapter $2$: Metric Spaces