Topology

Contents......Page 8
Preface......Page 6
Notation......Page 7
Introduction......Page 13
1. Open Sets, Closed Sets, Neighborhoods, Bounds of a Set......Page 14
2. Limit of a Sequence. The Cauchy Criterion for Convergence......Page 18
3. Compactness of Closed Bounded Intervals......Page 20
4. Topology of the Space R^n......Page 21
5. Open Sets, Closed Sets, Neighborhoods......Page 23
6. Closure, Interior, Boundary......Page 26
7. Continuous Functions. Homeomorphisms......Page 30
8. Notion of a Limit......Page 35
9. Subspaces of a Topological Space......Page 39
10. Finite Products of Spaces......Page 42
11. Compact Spaces......Page 46
12. Locally Compact Spaces; Compactification......Page 53
13. Connectivity......Page 57
14. Topological Groups, Rings, and Fields......Page 63
15. Metrics and Ecarts......Page 72
16. Topology of a Metric Space......Page 79
17. Uniform Continuity......Page 83
18. Compact Metric Spaces......Page 87
19. Connected Metric Spaces......Page 90
20. Cauchy Sequences and Complete Spaces......Page 92
21. Idea of the Method of Successive Approximations......Page 98
22. Pointwise Convergence and Uniform Convergence......Page 101
23. Equicontinuous Spaces of Functions......Page 110
24. Total Variation and Length......Page 114
The Line R and the Space R^n......Page 122
Topological Spaces......Page 123
Metric Spaces......Page 128
Definitions and Axioms......Page 137
Bibliography......Page 138
1. Order Relation on F(E, R) and on F(E, \\bar{R})......Page 139
2. Bounds of a Numerical Function......Page 140
3. Upper and Lower Envelopes of a Family of Functions......Page 141
II. Limit Notions Associated with Numerical Functions......Page 143
4. Limits Superior and Inferior of a Function along a Filter Base on E......Page 144
5. Limits Superior and Inferior of a Family, of Functions......Page 146
6. Operations on Continuous Functions......Page 147
III. Semicontinuous Numerical Functions......Page 148
7. Semicontinuity at a Point......Page 149
8. Functions, Lower Semicontinuous on the Entire Space......Page 150
10. Semicontinuous Functions on a Compact Space......Page 153
11. Semicontinuity of Length......Page 154
IV. The Stone-Weierstrass Theorem (Section 12)......Page 158
V. Functions Defined on an Interval of R......Page 162
13. Left and Right Limits......Page 163
14. Monotone Functions......Page 165
15. Theorems of Finite Increase......Page 166
16. Definition of Convex Functions. Immediate Properties......Page 169
17. Continuity and Differentiability of Convex Functions......Page 171
18. Criteria for Convexity......Page 173
19. Convex Functions on a Subset of a Vector Space......Page 175
20. The Mean Relative to a Monotone Function......Page 178
Numerical Functions Defined on an Arbitrary Set......Page 185
Semicontinuous Numerical Functions......Page 186
Stone-Weierstrass Theorem......Page 187
Convex Functions......Page 188
Means and Inequalities......Page 191
Definitions and Axioms......Page 193
Bibliography......Page 194
1. Definition and Elementary Properties of Topological Vector Spaces......Page 195
2. Topology Associated with a Family of Seminorms......Page 199
3. Classical Examples of Topological Vector Spaces......Page 208
II. Normed Spaces......Page 213
4. Topology Associated with a Norm ; Continuous Linear Mappings......Page 214
5. Stability of Isomorphisms......Page 221
6. Product of Normed Spaces; Continuous Multilinear Mappings......Page 224
7. Finite-Dimensional Normed Spaces......Page 226
8. Summable Families of Real Numbers......Page 228
9. Summable Families in Topological Groups and Normed Spaces......Page 236
10. Series; Comparison of Series and Summable Families......Page 244
11. Series and Summable Families of Functions......Page 251
12. Multipliable Families and Infinite Products of Complex Numbers......Page 255
13. Normed Algebras......Page 263
IV. Hubert Spaces......Page 271
14. Definition and Elementary Properties of Prehilbert Spaces......Page 272
15. Orthogonal Projection. Study of the Dual......Page 280
16. Orthogonal Systems......Page 287
17. Fourier Series and Orthogonal Polynomials......Page 294
General Topological Vector Spaces......Page 299
Topology Associated with a Family of Seminorms......Page 301
Topology Associated with a Norm......Page 304
Comparison of Norms......Page 305
Norms and Convex Functions......Page 306
Linear Functionals on Normed Spaces......Page 308
Topological Dual and Bidual......Page 309
Compact Linear Mappings......Page 310
Complete Normed Spaces......Page 312
Separable Normed Spaces......Page 314
Discontinuous Linear Mappings......Page 315
Finite-Dimensional Normed Spaces......Page 316
Summable Families of Real or Complex Numbers......Page 317
Summable Families in Topological Groups and Normed Spaces......Page 318
Series; Comparison of Series and Summable Families......Page 320
Summable Series and Families of Functions......Page 322
Multipliable Families and Infinite Products of Complex Numbers......Page 326
Normed Algebras......Page 328
Elementary Properties of Prehilbert Spaces......Page 329
Orthogonal Projection. Study of the Dual......Page 333
Orthogonal Systems......Page 338
Orthogonal Polynomials......Page 341
Definitions and Axioms......Page 343
Notation......Page 345
Bibliography......Page 346
Subject Index......Page 347