HANDBOOK
OF ANALYSIS AND ITS FOUNDATIONS
by Eric
Schechter
Detailed Table of Contents
Preface, xiii
-- About the Choice of Topics, xiii
-- Existence, Examples, and Intangibles, xv
-- Abstract versus Concrete, xviii
-- Order of Topics, xix
-- How to Use This Book, xx
-- Acknowledgments, xxi
-- To Contact Me, xxii
Part A: SETS AND ORDERINGS, 1
1. Sets, 3
-- Mathematical Language and Informal Logic, 3
-- Basic Notations for Sets, 11
-- Ways to Combine Sets, 15
-- Functions and Products of Sets, 19
-- ZF Set Theory, 25
2. Functions, 34
-- Some Special Functions, 34
-- Distances, 39
-- Cardinality, 43
-- Induction and Recursion on the Integers, 47
3. Relations and Orderings, 49
-- Relations, 50
-- Preordered Sets, 52
-- More about Equivalences, 54
-- More about Posets, 56
-- Max, Sup, and Other Special Elements, 59
-- Chains, 62
-- Van Maaren's Geometry-Free Sperner Lemma, 64
-- Well Ordered Sets, 72
4. More about Sups and Infs, 78
-- Moore Collections and Moore Closures, 78
-- Some Special Types of Moore Closures, 83
-- Lattices and Completeness, 87
-- More about Lattices, 88
-- More about Complete Lattices, 91
-- Order Completions, 92
-- Sups and Infs in Metric Spaces, 97
5. Filters, Topologies, and Other Sets of Sets, 100
-- Filters and Ideals, 100
-- Topologies, 106
-- Algebras and Sigma-Algebras, 115
-- Uniformities, 118
-- Images and Preimages of Sets of Sets, 122
-- Transitive Sets and Ordinals, 122
-- The Class of Ordinals, 127
6. Constructivism and Choice, 131
-- Examples of Nonconstructive Mathematics, 132
-- Further Comments on Constructivism, 135
-- The Meaning of Choice, 139
-- Variants and Consequences of Choice, 141
-- Some Equivalents of Choice, 144
-- Countable Choice, 148
-- Dependent Choice, 149
-- The Ultrafilter Principle, 150
7. Nets and Convergences, 155
-- Nets, 157
-- Subnets, 161
-- Universal Nets, 165
-- More about Subsequences, 167
-- Convergence Spaces, 168
-- Convergence in Posets, 171
-- Convergence in Complete Lattices, 174
Part B: ALGEBRA, 177
8. Elementary Algebraic Systems, 179
-- Monoids, 179
-- Groups, 181
-- Sums and Quotients of Groups, 184
-- Rings and Fields, 187
-- Matrices, 192
-- Ordered Groups, 194
-- Lattice Groups, 197
-- Universal Algebras, 202
-- Examples of Equational Varieties, 205
9. Concrete Categories, 208
-- Definitions and Axioms, 210
-- Examples of Categories, 212
-- Initial Structures and Other Categorical Constructions, 217
-- Varieties with Ideals, 221
-- Functors, 227
-- The Reduced Power Functor, 229
-- Exponential (Dual) Functors, 238
10. The Real Numbers, 242
-- Dedekind Completions of Ordered Groups, 242
-- Ordered Fields and the Reals, 245
-- The Hyperreal Numbers, 250
-- Quadratic Extensions and the Complex Numbers, 254
-- Absolute Values, 259
-- Convergence of Sequences and Series, 263
11. Linearity, 272
-- Linear Spaces and Linear Subspaces, 272
-- Linear Maps, 277
-- Linear Dependence, 280
-- Further Results in Finite Dimensions, 282
-- Choice and Vector Bases, 285
-- Dimension of the Linear Dual (Optional), 287
-- Preview of Measure and Integration, 288
-- Ordered Vector Spaces, 292
-- Positive Operators, 296
-- Orthogonality in Riesz Spaces (Optional), 300
12. Convexity, 302
-- Convex Sets, 302
-- Combinatorial Convexity in Finite Dimensions (Optional), 307
-- Convex Functions, 308
-- Norms, Balanced Functionals, and Other Special Functions, 313
-- Minkowski Functionals, 315
-- Hahn-Banach Theorems, 317
-- Convex Operators, 319
13. Boolean Algebras, 326
-- Boolean Lattices, 326
-- Boolean Homomorphisms and Subalgebras, 329
-- Boolean Rings, 334
-- Boolean Equivalents of UF, 338
-- Heyting Algebras, 340
14. Logic and Intangibles, 344
-- Some Informal Examples of Models, 345
-- Languages and Truths, 350
-- Ingredients of First-Order Language, 354
-- Assumptions in First-Order Logic, 362
-- Some Syntactic Results (Propositional Logic), 366
-- Some Syntactic Results (Predicate Logic), 372
-- The Semantic View, 377
-- Soundness, Completeness, and Compactness, 385
-- Nonstandard Analysis, 394
-- Summary of Some Consistency Results, 399
-- Quasiconstructivism and Intangibles, 403
Part C: TOPOLOGY AND UNIFORMITY, 407
15. Topological Spaces, 409
-- Pretopological Spaces, 409
-- Topological Spaces and Their Convergences, 411
-- More about Topological Closures, 415
-- Continuity, 417
-- More about Initial and Product Topologies, 421
-- Quotient Topologies, 425
-- Neighborhood Bases and Topology Bases, 426
-- Cluster Points, 430
-- More about Intervals, 431
16. Separation and Regularity Axioms, 435
-- Kolmogorov (T-Zero) Topologies and Quotients, 436
-- Symmetric and Frechet (T-One) Topologies, 438
-- Preregular and Hausdorff (T-Two) Topologies, 439
-- Regular and T-Three Topologies, 441
-- Completely Regular and Tychonov (T-Three and a Half) Topologies, 442
-- Partitions of Unity, 444
-- Normal Topologies, 446
-- Paracompactness, 448
-- Hereditary and Productive Properties, 451
17. Compactness, 453
-- Characterizations in Terms of Convergences, 453
-- Basic Properties of Compactness, 456
-- Regularity and Compactness, 458
-- Tychonov's Theorem, 461
-- Compactness and Choice (Optional), 461
-- Compactness, Maxima, and Sequences, 466
-- Pathological Examples: Ordinal Spaces (Optional), 472
-- Boolean Spaces, 473
-- Eberlein-Smulian Theorem, 477
18. Uniform Spaces, 481
-- Lipschitz Mappings, 482
-- Uniform Continuity, 484
-- Pseudometrizable Gauges, 487
-- Compactness and Uniformity, 490
-- Uniform Convergence, 491
-- Equicontinuity, 493
19. Metric and Uniform Completeness, 499
-- Cauchy Filters, Nets, and Sequences, 499
-- Complete Metrics and Uniformities, 502
-- Total Boundedness and Precompactness, 505
-- Bounded Variation, 508
-- Cauchy Continuity, 511
-- Cauchy Spaces (Optional), 512
-- Completions, 513
-- Banach's Fixed Point Theorem, 516
-- Meyers's Converse (Optional), 520
-- Bessaga's Converse and Bronsted's Principle (Optional), 523
20. Baire Theory, 530
-- G-Delta Sets, 530
-- Meager Sets, 531
-- Generic Continuity Theorems, 533
-- Topological Completeness, 536
-- Baire Spaces and the Baire Category Theorem, 537
-- Almost Open Sets, 539
-- Relativization, 540
-- Almost Homeomorphisms, 541
-- Tail Sets, 543
-- Baire Sets (Optional), 545
21. Positive Measure and Integration, 547
-- Measurable Functions, 547
-- Joint Measurability, 549
-- Positive Measures and Charges, 552
-- Null Sets, 554
-- Lebesgue Measure, 556
-- Some Countability Arguments, 559
-- Convergence in Measure, 561
-- Integration of Positive Functions, 565
-- Essential Suprema, 569
Part D: TOPOLOGICAL VECTOR SPACES, 573
22. Norms, 575
-- (G-)(Semi)Norms, 575
-- Basic Examples, 578
-- Sup Norms, 581
-- Convergent Series, 585
-- Bochner-Lebesgue Spaces, 589
-- Strict Convexity and Uniform Convexity, 596
-- Hilbert Spaces, 601
23. Normed Operators, 607
-- Norms of Operators, 607
-- Equicontinuity and Joint Continuity, 612
-- The Bochner Integral, 65
-- Hahn-Banach Theorems in Normed Spaces, 617
-- A Few Consequences of HB, 621
-- Duality and Separability, 622
-- Unconditionally Convergent Series, 624
-- Neumann Series and Spectral Radius (Optional), 627
24. Generalized Riemann Integrals, 629
-- Definitions of the Integrals, 629
-- Basic Properties of Gauge Integrals, 635
-- Additivity over Partitions, 638
-- Integrals of Continuous Functions, 642
-- Monotone Convergence Theorem, 645
-- Absolute Integrability, 647
-- Henstock and Lebesgue Integrals, 649
-- More about Lebesgue Measure, 656
-- More about Riemann Integrals (Optional), 658
25. Frechet Derivatives, 661
-- Definitions and Basic Properties, 661
-- Partial Derivatives, 665
-- Strong Derivatives, 669
-- Derivatives of Integrals, 674
-- Integrals of Derivatives, 675
-- Some Applications of the Second Fundamental Theorem of Calculus, 677
-- Path Integrals and Analytic Functions (Optional), 683
26. Metrization of Groups and Vector Spaces, 688
-- F-Seminorms, 689
-- TAG's and TVS's, 697
-- Arithmetic in TAG's and TVS's, 700
-- Neighborhoods of Zero, 702
-- Characterizations in Terms of Gauges, 705
-- Uniform Structure of TAG's, 708
-- Pontryagin Duality and Haar Measure (Optional; Proofs Omitted), 710
-- Ordered Topological Vector Spaces, 714
27. Barrels and Other Features of TVS's, 721
-- Bounded Subsets of TVS's, 721
-- Bounded Sets in Ordered TVS's, 726
-- Dimension in TVS's, 728
-- Fixed Point Theorems of Brouwer, Schauder, and Tychonov, 730
-- Barrels and Ultrabarrels, 732
-- Proofs of Barrel Theorems, 736
-- Inductive Topologies and LF Spaces, 744
-- The Dream Universe of Garnir and Wright, 748
28. Duality and Weak Compactness, 752
-- Hahn-Banach Theorems in TVS's, 752
-- Bilinear Pairings, 754
-- Weak Topologies, 758
-- Weak Topologies of Normed Spaces, 761
-- Polar Arithmetic and Equicontinuous Sets, 764
-- Duals of Product Spaces, 769
-- Characterizations of Weak Compactness, 771
-- Some Consequences in Banach Spaces, 777
-- More about Uniform Convexity, 780
-- Duals of the Lebesgue Spaces, 782
29. Vector Measures, 785
-- Basic Properties, 785
-- The Variation of a Charge, 787
-- Indefinite Bochner Integrals and Radon-Nikodym Derivatives, 790
-- Conditional Expectations and Martingales, 792
-- Existence of Radon-Nikodym Derivatives, 796
-- Semivariation and Bartle Integrals, 802
-- Measures on Intervals, 806
-- Pincus's Pathology (Optional), 810
30. Initial Value Problems, 814
-- Elementary Pathological Examples, 815
-- Caratheodory Solutions, 816
-- Lipschitz Conditions, 819
-- Generic Solvability, 822
-- Compactness Conditions, 822
-- Isotonicity Conditions, 824
-- Generalized Solutions, 826
-- Semigroups and Dissipative Operators, 828
References, 839
Index and Symbol List, 857