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Introduction to Measure Theory and Functional Analysis 2015 ed. [Minkštas viršelis]

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  • Formatas: Paperback / softback, 314 pages, aukštis x plotis: 235x155 mm, weight: 4978 g, 1 Illustrations, color; 7 Illustrations, black and white; XIV, 314 p. 8 illus., 1 illus. in color., 1 Paperback / softback
  • Serija: UNITEXT 89
  • Išleidimo metai: 13-May-2015
  • Leidėjas: Springer International Publishing AG
  • ISBN-10: 331917018X
  • ISBN-13: 9783319170183
Kitos knygos pagal šią temą:
Introduction to Measure Theory and Functional Analysis 2015 ed.Didesnis paveikslėlis
  • Formatas: Paperback / softback, 314 pages, aukštis x plotis: 235x155 mm, weight: 4978 g, 1 Illustrations, color; 7 Illustrations, black and white; XIV, 314 p. 8 illus., 1 illus. in color., 1 Paperback / softback
  • Serija: UNITEXT 89
  • Išleidimo metai: 13-May-2015
  • Leidėjas: Springer International Publishing AG
  • ISBN-10: 331917018X
  • ISBN-13: 9783319170183
Kitos knygos pagal šią temą:
Pastovi nuoroda: https://www.kriso.lt/db/9783319170183.html
Raktažodžiai:

This book introduces readers to theories that play a crucial role in modern mathematics, such as integration and functional analysis, employing a unifying approach that views these two subjects as being deeply intertwined. This feature is particularly evident in the broad range of problems examined, the solutions of which are often supported by generous hints. If the material is split into two courses, it can be supplemented by additional topics from the third part of the book, such as functions of bounded variation, absolutely continuous functions, and signed measures.

This textbook addresses the needs of graduate students in mathematics, who will find the basic material they will need in their future careers, as well as those of researchers, who will appreciate the self-contained exposition which requires no other preliminaries than basic calculus and linear algebra.

Recenzijos

This is an excellent introductory text on measure theory and integration, with a very good presentation of two fundamental aspects of functional analysis: Hilbert spaces and Banach spaces. The material is presented at a level accessible for a graduate student in mathematics or a researcher in another, related discipline. Each section contains a large number of examples and exercises accompanied by generous hints. (Victoria R. Steblovskaya, Mathematical Reviews, June, 2016)

This book, written by leading experts, is a well-crafted textbook covering a medley of relevant topics in measure theory and functional analysis in a rather get-to-the-point-quickly fashion, yet resulting in a very readable and enjoyable journey. The book manages to cover quite a lot of ground in a relatively short number of pages strewn with plenty of exercises. All-in-all, the student reading this book stands to gain much knowledge, develop and hone her skills, and enjoy the trip. (Ittay Weiss, MAA Reviews, maa.org, April, 2016)

Part I Measure and Integration
1 Measure Spaces
3(34)
1.1 Algebras and σ-Algebras of Sets
3(4)
1.1.1 Notation and Preliminaries
3(2)
1.1.2 Algebras and σ-Algebras
5(2)
1.2 Measures
7(6)
1.2.1 Additive and σ-Additive Functions
7(3)
1.2.2 Measure Spaces
10(2)
1.2.3 Borel--Cantelli Lemma
12(1)
1.3 The Basic Extension Theorem
13(7)
1.3.1 Monotone Classes
13(3)
1.3.2 Outer Measures
16(4)
1.4 Borel Measures on RN
20(17)
1.4.1 Lebesgue Measure on [ 0, 1)
20(2)
1.4.2 Lebesgue Measure on R
22(3)
1.4.3 Lebesgue Measure on RN
25(2)
1.4.4 Examples
27(2)
1.4.5 Regularity of Radon Measures
29(6)
References
35(2)
2 Integration
37(44)
2.1 Measurable Functions
38(6)
2.1.1 Inverse Image of a Function
38(1)
2.1.2 Measurable Maps and Borel Functions
38(6)
2.2 Convergence Almost Everywhere
44(2)
2.3 Approximation by Continuous Functions
46(3)
2.4 Integral of Borel Functions
49(18)
2.4.1 Integral of Positive Simple Functions
50(1)
2.4.2 Repartition Function
51(2)
2.4.3 The Archimedean Integral
53(3)
2.4.4 Integral of Positive Borel Functions
56(6)
2.4.5 Integral of Functions with Variable Sign
62(5)
2.5 Convergence of Integrals
67(11)
2.5.1 Dominated Convergence
67(4)
2.5.2 Uniform Summability
71(3)
2.5.3 Integrals Depending on a Parameter
74(4)
2.6 Miscellaneous Exercises
78(3)
3 LP Spaces
81(26)
3.1 The Spaces lp(X, μ) and LP(X, μ)
81(8)
3.2 The Space L∞ (X, μ)
89(5)
3.3 Convergence in Measure
94(1)
3.4 Convergence and Approximation in LP
95(8)
3.4.1 Convergence Results
95(3)
3.4.2 Dense Subsets in LP
98(5)
3.5 Miscellaneous Exercises
103(4)
References
106(1)
4 Product Measures
107(26)
4.1 Product Spaces
107(8)
4.1.1 Product Measures
107(5)
4.1.2 Fubini-Tonelli Theorem
112(3)
4.2 Compactness in LP
115(3)
4.3 Convolution and Approximation
118(15)
4.3.1 Convolution Product
119(4)
4.3.2 Approximation by Smooth Functions
123(7)
References
130(3)
Part II Functional Analysis
5 Hilbert Spaces
133(34)
5.1 Definitions and Examples
134(4)
5.2 Orthogonal Projection
138(7)
5.2.1 Projection onto a Closed Convex Set
139(2)
5.2.2 Projection onto a Closed Subspace
141(4)
5.3 Riesz Representation Theorem
145(8)
5.3.1 Bounded Linear Functionals
145(2)
5.3.2 Riesz Theorem
147(6)
5.4 Orthonormal Sequences and Bases
153(10)
5.4.1 Bessel's Inequality
154(1)
5.4.2 Orthonormal Bases
155(4)
5.4.3 Completeness of the Trigonometric System
159(4)
5.5 Miscellaneous Exercises
163(4)
References
166(1)
6 Banach Spaces
167(62)
6.1 Definitions and Examples
168(2)
6.2 Bounded Linear Operators
170(14)
6.2.1 The Principle of Uniform Boundedness
176(2)
6.2.2 The Open Mapping Theorem
178(6)
6.3 Bounded Linear Functionals
184(16)
6.3.1 Hahn-Banach Theorem
185(5)
6.3.2 Separation of Convex Sets
190(4)
6.3.3 The Dual of lP
194(6)
6.4 Weak Convergence and Reflexivity
200(15)
6.4.1 Reflexive Spaces
201(3)
6.4.2 Weak Convergence and Bolzano-Weierstrass Property
204(11)
6.5 Miscellaneous Exercises
215(14)
References
225(4)
Part III Selected Topics
7 Absolutely Continuous Functions
229(24)
7.1 Monotone Functions
230(7)
7.1.1 Differentiation of Monotone Functions
231(6)
7.2 Functions of Bounded Variation
237(5)
7.3 Absolutely Continuous Functions
242(9)
7.4 Miscellaneous Exercises
251(2)
8 Signed Measures
253(18)
8.1 Comparison Between Measures
254(2)
8.2 Lebesgue Decomposition
256(5)
8.2.1 The Case of Finite Measures
256(3)
8.2.2 The General Case
259(2)
8.3 Signed Measures
261(6)
8.3.1 Total Variation
261(3)
8.3.2 Radon-Nikodym Theorem
264(1)
8.3.3 Hahn Decomposition
265(2)
8.4 Dual of LP (X, μ)
267(4)
Reference
270(1)
9 Set-Valued Functions
271(8)
9.1 Definitions and Examples
271(2)
9.2 Existence of a Summable Selection
273(6)
Reference
277(2)
Appendix A Distance Function 279(6)
Appendix B Semicontinuous Functions 285(4)
Appendix C Finite-Dimensional Linear Spaces 289(4)
Appendix D Baire's Lemma 293(2)
Appendix E Relatively Compact Families of Continuous Functions 295(4)
Appendix F Legendre Transform 299(4)
Appendix G Vitali's Covering Theorem 303(4)
Appendix H Ekeland's Variational Principle 307(4)
Index 311
Prof. Piermarco Cannarsa, Full professor in Mathematical Analysis, Dept. of Mathematics, Universitą degli Studi di Roma "Tor Vergata", via della Ricerca Scientifica 1, 00133 Roma, Italy and Italian coordinator of the European Research Group (GDRE) on "Control of Partial Differential Equations" (CONEDP) issued by CNRS, INdAM and Universite' de Provence.

Prof. Teresa D'Aprile, Dept. of Mathematics, Universitą degli Studi di Roma "Tor Vergata", via della Ricerca Scientifica 1, 00133 Roma, Italy