Handbook of analysis and its foundations

Cover......Page 1
Date-line......Page 2
Contents......Page 3
About the Choice of Topics......Page 9
Existence, Examples, and Intangibles......Page 11
Abstract versus Concrete......Page 14
Order of Topics......Page 15
How to Use This Book......Page 16
Acknowledgments......Page 17
To Contact Me......Page 18
A SETS AND ORDERINGS......Page 19
Mathematical Language and Informal Logic......Page 21
Basic Notations for Sets......Page 29
Ways to Combine Sets......Page 33
Functions and Products of Sets......Page 37
ZF Set Theory......Page 43
Some Special Functions......Page 52
Distances......Page 57
Cardinality......Page 61
Induction and Recursion on the Integers......Page 65
3 Relations and Orderings......Page 67
Relations......Page 68
Preordered Sets......Page 70
More about Equivalences......Page 72
More about Posets......Page 74
Max, Sup, and Other Special Elements......Page 77
Chains......Page 80
Van Maaren\'s Geometry-Free Sperner Lemma......Page 82
Well Ordered Sets......Page 90
Moore Collections and Moore Closures......Page 96
Some Special Types of Moore Closures......Page 101
Lattices and Completeness......Page 105
More about lattices......Page 106
More about Complete lattices......Page 109
Order Completions......Page 110
Sups and Infs in Metric Spaces......Page 115
Filters and Ideals......Page 118
Topologies......Page 124
Algebras and Sigma- Algebras......Page 133
Uniformities......Page 136
Transitive Sets and Ordinals......Page 140
The Class of Ordinals......Page 145
6 Constructivism and Choice......Page 149
Examples of Nonconstructive Mathematics......Page 150
Further Comments on Constructivism......Page 153
The Meaning of Choice......Page 157
Variants and Consequences of Choice......Page 159
Some Equivalents of Choice......Page 162
Countable Choice......Page 166
Dependent Choice......Page 167
The Ultrafilter Principle......Page 168
7 Nets and Convergences......Page 173
Nets......Page 175
Subnets......Page 179
Universal Nets......Page 183
More about Subsequences......Page 185
Convergence Spaces......Page 186
Convergence in Posets......Page 189
Convergence in Complete Lattices......Page 192
B ALGEBRA......Page 195
Monoids......Page 197
Groups......Page 199
Sums and Quotients of Groups......Page 202
Rings and Fields......Page 205
Matrices......Page 210
Ordered Groups......Page 212
Lattice Groups......Page 215
Universal Algebras......Page 220
Examples of Equational Varieties......Page 223
9 Concrete Categories......Page 226
Definitions and Axioms......Page 228
Examples of Categories......Page 230
Initial Structures and Other Categorical Constructions......Page 235
Varieties with Ideals......Page 239
Functors......Page 245
The Reduced Power Functor......Page 247
Exponential (Dual) Functors......Page 256
Dedekind Completions of Ordered Groups......Page 260
Ordered Fields and the Reals......Page 263
The Hyperreal Numbers......Page 268
Quadratic Extensions and the Complex Numbers......Page 272
Absolute Values......Page 277
Convergence of Sequences and Series......Page 281
Linear Spaces and Linear Subspaccs......Page 290
Linear Maps......Page 295
Linear Dependence......Page 298
Further Results in Finite Dimensions......Page 300
Choice and Vector Bases......Page 303
Dimension of the Linear Dual (Optional)......Page 305
Preview of Measure and Integration......Page 306
Ordered Vector Spaces......Page 310
Positive Operators......Page 314
Orthogonality in Riesz Spaces (Optional)......Page 318
Convex Sets......Page 320
Combinatorial Convexity in Finite Dimensions (Optional)......Page 325
Convex Functions......Page 326
Norms, Balanced Functionals, and Other Special Functions......Page 331
Minkowski Funclionals......Page 333
Hahn-Banach Theorems......Page 335
Convex Operators......Page 337
Boolean Lattices......Page 344
Boolean Homomorphisms and Subalgebras......Page 347
Boolean Rings......Page 352
Boolean Equivalents of UF......Page 356
Heyting Algebras......Page 358
14 Logic and Intangibles......Page 362
Some Informal Examples of Models......Page 363
Languages and Truths......Page 368
Ingredients of First-Order Language......Page 372
Assumptions in First-Order Logic......Page 380
Some Syntactic Results (Propositional Logic)......Page 384
Some Syntactic Results (Predicate Logic)......Page 390
The Semantic View......Page 395
Soundness, Completeness, and Compactness......Page 403
Nonstandard Analysis......Page 412
Summary of Some Consistency Results......Page 417
Quasiconstructivism and Intangibles......Page 421
C TOPOLOGY AND UNIFORMITY......Page 425
Pretopological Spaces......Page 427
Topological Spaces and Their Convergences......Page 429
More about Topological Closures......Page 433
Continuity......Page 435
More about Initial and Product Topologies......Page 439
Quotient Topologies......Page 443
Neighborhood Bases and Topology Bases......Page 444
Cluster Points......Page 448
More about Intervals......Page 449
16 Separation and Regularity Axioms......Page 453
Kolmogorov (T-Zero) Topologies and Quotients......Page 454
Symmetric and Frechet (T-One) Topologies......Page 456
Preregular and Hausdorff (T-Two) Topologies......Page 457
Regular and T-Three Topologies......Page 459
Completely Regular and Tychonov (T-Three and a Half) Topologies......Page 460
Partitions of Unity......Page 462
Normal Topologies......Page 464
Paracompactness......Page 466
Hereditary and Productive Properties......Page 469
Characterizations in Terms of Convergences......Page 471
Basic Properties of Compactness......Page 474
Regularity and Compactness......Page 476
Compactness and Choice (Optional)......Page 479
Compactness, Maxima, and Sequences......Page 484
Pathological Examples: Ordinal Spaces (Optional)......Page 490
Boolean Spaces......Page 491
Eberlein-Smulian Theorem......Page 495
18 Uniform Spaces......Page 499
Lipschitz Mappings......Page 500
Uniform Continuity......Page 502
Pseudometrizable Gauges......Page 505
Compactness and Uniformity......Page 508
Uniform Convergence......Page 509
Equicontinuity......Page 511
Cauchy Filters, Nets, and Sequences......Page 517
Complete Metrics and Uniformities......Page 520
Total Boundedness and Precompactness......Page 523
Bounded Variation......Page 526
Cauchy Continuity......Page 529
Cauchy Spaces (Optional)......Page 530
Completions......Page 531
Banach\'s Fixed Point Theorem......Page 534
Meyers\'s Converse (Optional)......Page 538
Bessaga\'s Converse and Broensted\'s Principle (Optional)......Page 541
G-Delta Sets......Page 548
Meager Sets......Page 549
Generic Continuity Theorems......Page 551
Topological Completeness......Page 554
Baire Spaces and the Baire Category Theorem......Page 555
Almost Open Sets......Page 557
Relativization......Page 558
Almost Homeomorphisms......Page 559
Tail Sets......Page 561
Baire Sets (Optional)......Page 563
Measurable Functions......Page 565
Joint Measurability......Page 567
Positive Measures and Charges......Page 570
Null Sets......Page 572
Lebesgue Measure......Page 574
Some Countability Arguments......Page 577
Convergence in Measure......Page 579
Integration of Positive Functions......Page 583
Essential Suprema......Page 587
D TOPOLOGICAL VECTOR SPACES......Page 591
G-)(Semi)Norms......Page 593
Basic Examples......Page 596
Sup Norms......Page 599
Convergent Series......Page 603
Bochner-Lebesgue Spaces......Page 607
Strict Convexity and Uniform Convexity......Page 614
Hilbert Spaces......Page 619
Norms of Operators......Page 625
Equicontinuity and Joint Continuity......Page 630
The Bochner Integral......Page 633
Hahn-Banach Theorems in Normed Spaces......Page 635
A Few Consequences of HB......Page 639
Duality and Separability......Page 640
Unconditionally Convergent Series......Page 642
Neumann Series and Spectral Radius (Optional)......Page 645
Definitions of the Integrals......Page 647
Basic Properties of Gauge Integrals......Page 653
Additivity over Partitions......Page 656
Integrals of Continuous Functions......Page 660
Monotone Convergence Theorem......Page 663
Absolute Integrability......Page 665
Henstock and Lebesgue Integrals......Page 667
More about Lebesgue Measure......Page 674
More about Riemann Integrals (Optional)......Page 676
Definitions and Basic Properties......Page 679
Partial Derivatives......Page 683
Strong Derivatives......Page 687
Derivatives of Integrals......Page 692
Integrals of Derivatives......Page 693
Some Applications of the Second Fundamental Theorem of Calculus......Page 695
Path Integrals and Analytic Functions (Optional)......Page 701
26 Metrization of Groups and Vector Spaces......Page 706
P-Seminorms......Page 707
TAG\'s and TVS\'s......Page 715
Arithmetic in TAG\'s and TVS\'s......Page 718
Neighborhoods of Zero......Page 720
Characterizations in Terms of Gauges......Page 723
Uniform Structure of TAG\'s......Page 726
Pontryagin Duality and Haar Measure (Optional; Proofs Omitted)......Page 728
Ordered Topological Vector Spaces......Page 732
Bounded Subsets of TVS\'s......Page 739
Bounded Sets in Ordered TVS\'s......Page 744
Dimension in TVS\'s......Page 746
Fixed Point Theorems of Brouwer, Schauder, and Tychonov......Page 748
Barrels and Ultrabarrels......Page 750
Proofs of Barrel Theorems......Page 754
Inductive Topologies and LF Spaces......Page 762
The Dream Universe of Garnir and Wright......Page 766
Hahn-Banach Theorems in TVS\'s......Page 770
Bilinear Pairings......Page 772
Weak Topologies......Page 776
Weak Topologies of Normed Spaces......Page 779
Polar Arithmetic and Equicontiiiuous Sets......Page 782
Duals of Product Spaces......Page 787
Characterizations of Weak Compactness......Page 789
Some Consequences in Banach Spaces......Page 795
More about Uniform Convexity......Page 798
Duals of the Lebesgue Spares......Page 800
Basic Properties......Page 803
The Variation of a Charge......Page 805
Indefinite Bochner Integrals and Radon-Nikodym Derivatives......Page 808
Conditional Expectations and Martingales......Page 810
Existence of Radon-Nikodym Derivatives......Page 814
Semivariation and Bartle Integrals......Page 820
Measures on Intervals......Page 824
Pincus\'s Pathology (Optional)......Page 828
30 Initial Value Problems......Page 832
Elementary Pathological Examples......Page 833
Caratheodory Solutions......Page 834
Lipschitz Conditions......Page 837
Compactness Conditions......Page 840
Isotonicity Conditions......Page 842
Generalized Solutions......Page 844
Semigroups and Dissipative Operators......Page 846
References......Page 857
Index and Symbol List......Page 875