Handbook of analysis and its foundations:
Saved in:
| Main Author: | |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
San Diego [u.a.]
Acad. Press
1997
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| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Physical Description: | XXII, 883 S. |
| ISBN: | 0126227608 |
Staff View
MARC
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| 100 | 1 | |a Schechter, Eric |e Verfasser |4 aut | |
| 245 | 1 | 0 | |a Handbook of analysis and its foundations |c Eric Schechter |
| 264 | 1 | |a San Diego [u.a.] |b Acad. Press |c 1997 | |
| 300 | |a XXII, 883 S. | ||
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Record in the Search Index
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| adam_text | 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
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
vii
viii Contents
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
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 Functional,
and Other Special Functions, 313 • Minkowski Functionate, 315 • Hahn
Banach Theorems, 317 • Convex Operators, 319
Contents ix
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
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
x Contents
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
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, 615 • 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
xi
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
|
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| spelling | Schechter, Eric Verfasser aut Handbook of analysis and its foundations Eric Schechter San Diego [u.a.] Acad. Press 1997 XXII, 883 S. txt rdacontent n rdamedia nc rdacarrier Mathematical analysis Analysis (DE-588)4001865-9 gnd rswk-swf Analysis (DE-588)4001865-9 s DE-604 HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=007687768&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Schechter, Eric Handbook of analysis and its foundations Mathematical analysis Analysis (DE-588)4001865-9 gnd |
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| title | Handbook of analysis and its foundations |
| title_auth | Handbook of analysis and its foundations |
| title_exact_search | Handbook of analysis and its foundations |
| title_full | Handbook of analysis and its foundations Eric Schechter |
| title_fullStr | Handbook of analysis and its foundations Eric Schechter |
| title_full_unstemmed | Handbook of analysis and its foundations Eric Schechter |
| title_short | Handbook of analysis and its foundations |
| title_sort | handbook of analysis and its foundations |
| topic | Mathematical analysis Analysis (DE-588)4001865-9 gnd |
| topic_facet | Mathematical analysis Analysis |
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