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Introduction to Real Analysis 2019 ed. [Kietas viršelis]

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  • Formatas: Hardback, 386 pages, aukštis x plotis: 235x155 mm, weight: 793 g, 1 Illustrations, black and white; XXXII, 386 p. 1 illus., 1 Hardback
  • Serija: Graduate Texts in Mathematics 280
  • Išleidimo metai: 30-Jul-2019
  • Leidėjas: Springer Nature Switzerland AG
  • ISBN-10: 3030269019
  • ISBN-13: 9783030269012
Kitos knygos pagal šią temą:
Introduction to Real Analysis 2019 ed.Didesnis paveikslėlis
  • Formatas: Hardback, 386 pages, aukštis x plotis: 235x155 mm, weight: 793 g, 1 Illustrations, black and white; XXXII, 386 p. 1 illus., 1 Hardback
  • Serija: Graduate Texts in Mathematics 280
  • Išleidimo metai: 30-Jul-2019
  • Leidėjas: Springer Nature Switzerland AG
  • ISBN-10: 3030269019
  • ISBN-13: 9783030269012
Kitos knygos pagal šią temą:
Pastovi nuoroda: https://www.kriso.lt/db/9783030269012.html
Raktažodžiai:

Developed over years of classroom use, this textbook provides a clear and accessible approach to real analysis. This modern interpretation is based on the author’s lecture notes and has been meticulously tailored to motivate students and inspire readers to explore the material, and to continue exploring even after they have finished the book. The definitions, theorems, and proofs contained within are presented with mathematical rigor, but conveyed in an accessible manner and with language and motivation meant for students who have not taken a previous course on this subject.

The text covers all of the topics essential for an introductory course, including Lebesgue measure, measurable functions, Lebesgue integrals, differentiation, absolute continuity, Banach and Hilbert spaces, and more. Throughout each chapter, challenging exercises are presented, and the end of each section includes additional problems. Such an inclusive approach creates an abundance of opportunities for readers to develop their understanding, and aids instructors as they plan their coursework. Additional resources are available online, including expanded chapters, enrichment exercises, a detailed course outline, and much more.

Introduction to Real Analysis is intended for first-year graduate students taking a first course in real analysis, as well as for instructors seeking detailed lecture material with structure and accessibility in mind. Additionally, its content is appropriate for Ph.D. students in any scientific or engineering discipline who have taken a standard upper-level undergraduate real analysis course.

Recenzijos

This book is written in a clear style that is suitable for students reading on their own or as part of a guided class. this book gives an accessible introduction to real analysis with emphasis on Lebesgue measure and Lebesgue integration in Euclidean spaces. This book could be suitable as a primary text for a first course focused on measure theory in Euclidean spaces or, due to the numerous exercises throughout, as a supplemental text for instructors giving other introductory measure theory courses. (Gareth Speight, Mathematical Reviews, June, 2020) This book is intended primarily for students beginning their graduate studies in mathematics but it will also be suitable for well-prepared undergraduates. (Frédéric Morneau-Guérin, MAA Reviews, February 16, 2020) The book isreally a textbook full of intermediate motivated questions addressed to the audience and step-divided discussions. It can be suitable for first-year students in mathematics, for well-prepared undergraduate mathematical majors, and for graduate students from a variety of engineering and scientific applications. (Sergei V. Rogosin, zbMATH 1426.26001, 2020)

Preliminaries 1(14)
1 Metric and Normed Spaces
15(18)
1.1 Metric Spaces
15(7)
1.1.1 Convergence and Completeness
16(1)
1.1.2 Topology in Metric Spaces
17(1)
1.1.3 Compact Sets in Metric Spaces
18(2)
1.1.4 Continuity for Functions on Metric Spaces
20(2)
1.2 Normed Spaces
22(5)
1.2.1 Vector Spaces
22(1)
1.2.2 Seminorms and Norms
23(1)
1.2.3 Infinite Series in Normed Spaces
24(2)
1.2.4 Equivalent Norms
26(1)
1.3 The Uniform Norm
27(4)
1.3.1 Some Function Spaces
28(3)
1.4 Holder and Lipschitz Continuity
31(2)
2 Lebesgue Measure
33(54)
2.1 Exterior Lebesgue Measure
34(18)
2.1.1 Boxes
35(1)
2.1.2 Some Facts about Boxes
36(3)
2.1.3 Exterior Lebesgue Measure
39(5)
2.1.4 The Exterior Measure of a Box
44(2)
2.1.5 The Cantor Set
46(3)
2.1.6 Regularity of Exterior Measure
49(3)
2.2 Lebesgue Measure
52(19)
2.2.1 Definition and Basic Properties
53(2)
2.2.2 Toward Countable Additivity and Closure under Complements
55(4)
2.2.3 Countable Additivity
59(3)
2.2.4 Equivalent Formulations of Measurability
62(2)
2.2.5 Caratheodory's Criterion
64(2)
2.2.6 Almost Everywhere and the Essential Supremum
66(5)
2.3 More Properties of Lebesgue Measure
71(10)
2.3.1 Continuity from Above and Below
71(3)
2.3.2 Cartesian Products
74(1)
2.3.3 Linear Changes of Variable
75(6)
2.4 Nonmeasurable Sets
81(6)
2.4.1 The Axiom of Choice
81(1)
2.4.2 Existence of a Nonmeasurable Set
82(2)
2.4.3 Further Results
84(3)
3 Measurable Functions
87(32)
3.1 Definition and Properties of Measurable Functions
87(6)
3.1.1 Extended Real-Valued Functions
88(4)
3.1.2 Complex-Valued Functions
92(1)
3.2 Operations on Functions
93(10)
3.2.1 Sums and Products
93(2)
3.2.2 Compositions
95(1)
3.2.3 Suprema and Limits
96(2)
3.2.4 Simple Functions
98(5)
3.3 The Lebesgue Space L∞(E)
103(4)
3.3.1 Convergence and Completeness in L∞(E)
105(2)
3.4 Egorov's Theorem
107(4)
3.5 Convergence in Measure
111(6)
3.6 Luzin's Theorem
117(2)
4 The Lebesgue Integral
119(58)
4.1 The Lebesgue Integral of Nonnegative Functions
119(7)
4.1.1 Integration of Nonnegative Simple Functions
121(2)
4.1.2 Integration of Nonnegative Functions
123(3)
4.2 The Monotone Convergence Theorem and Fatou's Lemma
126(6)
4.2.1 The Monotone Convergence Theorem
127(3)
4.2.2 Fatou's Lemma
130(2)
4.3 The Lebesgue Integral of Measurable Functions
132(6)
4.3.1 Extended Real-Valued Functions
133(2)
4.3.2 Complex-Valued Functions
135(1)
4.3.3 Properties of the Integral
136(2)
4.4 Integrable Functions and L1(E)
138(8)
4.4.1 The Lebesgue Space L1(E)
138(2)
4.4.2 Convergence in L1-Norm
140(2)
4.4.3 Linearity of the Integral for Integrable Functions
142(1)
4.4.4 Inclusions between L1(E) and L∞(E)
143(3)
4.5 The Dominated Convergence Theorem
146(15)
4.5.1 The Dominated Convergence Theorem
147(2)
4.5.2 First Applications of the DCT
149(1)
4.5.3 Approximation by Continuous Functions
150(3)
4.5.4 Approximation by Really Simple Functions
153(1)
4.5.5 Relation to the Riemann Integral
154(7)
4.6 Repeated Integration
161(16)
4.6.1 Fubini's Theorem
161(7)
4.6.2 Tonelli's Theorem
168(3)
4.6.3 Convolution
171(6)
5 Differentiation
177(42)
5.1 The Cantor-Lebesgue Function
178(4)
5.2 Functions of Bounded Variation
182(12)
5.2.1 Definition and Examples
183(3)
5.2.2 Lipschitz and Holder Continuous Functions
186(1)
5.2.3 Indefinite Integrals and Antiderivatives
187(2)
5.2.4 The Jordan Decomposition
189(5)
5.3 Covering Lemmas
194(6)
5.3.1 The Simple Vitali Lemma
195(1)
5.3.2 The Vitali Covering Lemma
196(4)
5.4 Differentiability of Monotone Functions
200(7)
5.5 The Lebesgue Differentiation Theorem
207(12)
5.5.1 L1-Convergence of Averages
208(2)
5.5.2 Locally Integrable Functions
210(1)
5.5.3 The Maximal Theorem
210(3)
5.5.4 The Lebesgue Differentiation Theorem
213(3)
5.5.5 Lebesgue Points
216(3)
6 Absolute Continuity and the Fundamental Theorem of Calculus
219(34)
6.1 Absolutely Continuous Functions
220(3)
6.1.1 Differentiability of Absolutely Continuous Functions
222(1)
6.2 Growth Lemmas
223(6)
6.3 The Banach--Zaretsky Theorem
229(5)
6.4 The Fundamental Theorem of Calculus
234(7)
6.4.1 Applications of the FTC
235(2)
6.4.2 Integration by Parts
237(4)
6.5 The Chain Rule and Changes of Variable
241(4)
6.6 Convex Functions and Jensen's Inequality
245(8)
7 The Lp Spaces
253(36)
7.1 The lp Spaces
254(15)
7.1.1 Holder's Inequality
256(3)
7.1.2 Minkowski's Inequality
259(3)
7.1.3 Convergence in the lp Spaces
262(2)
7.1.4 Completeness of the lp Spaces
264(1)
7.1.5 Lp for p < 1
265(1)
7.1.6 c0 and c00
266(3)
7.2 The Lebesgue Space Lp(E)
269(8)
7.2.1 Seminorm Properties of || · ||p
271(1)
7.2.2 Identifying Functions That Are Equal Almost Everywhere
272(1)
7.2.3 Lp(E) for 0 < p < 1
273(1)
7.2.4 The Converse of Holder's Inequality
274(3)
7.3 Convergence in Lp-norm
277(7)
7.3.1 Dense Subsets of Lp(E)
279(5)
7.4 Separability of Lp(E)
284(5)
8 Hilbert Spaces and L2(E)
289(38)
8.1 Inner Products and Hilbert Spaces
289(6)
8.1.1 The Definition of an Inner Product
290(1)
8.1.2 Properties of an Inner Product
290(2)
8.1.3 Hilbert Spaces
292(3)
8.2 Orthogonality
295(10)
8.2.1 Orthogonal Complements
296(2)
8.2.2 Orthogonal Projections
298(4)
8.2.3 Characterizations of the Orthogonal Projection
302(1)
8.2.4 The Closed Span
302(1)
8.2.5 The Complement of the Complement
303(1)
8.2.6 Complete Sequences
304(1)
8.3 Orthonormal Sequences and Orthonormal Bases
305(15)
8.3.1 Orthonormal Sequences
305(2)
8.3.2 Unconditional Convergence
307(1)
8.3.3 Orthogonal Projections Revisited
308(2)
8.3.4 Orthonormal Bases
310(2)
8.3.5 Existence of an Orthonormal Basis
312(1)
8.3.6 The Legendre Polynomials
313(1)
8.3.7 The Haar System
314(2)
8.3.8 Unitary Operators
316(4)
8.4 The Trigonometric System
320(7)
9 Convolution and the Fourier Transform
327(60)
9.1 Convolution
327(17)
9.1.1 The Definition of Convolution
328(1)
9.1.2 Existence
329(2)
9.1.3 Convolution as Averaging
331(2)
9.1.4 Approximate Identities
333(5)
9.1.5 Young's Inequality
338(6)
9.2 The Fourier Transform
344(16)
9.2.1 The Inversion Formula
348(5)
9.2.2 Smoothness and Decay
353(7)
9.3 Fourier Series
360(18)
9.3.1 Periodic Functions
361(1)
9.3.2 Decay of Fourier Coefficients
362(3)
9.3.3 Convolution of Periodic Functions
365(1)
9.3.4 Approximate Identities and the Inversion Formula
365(6)
9.3.5 Completeness of the Trigonometric System
371(3)
9.3.6 Convergence of Fourier Series for p ≠ 2
374(4)
9.4 The Fourier Transform on L2(R)
378(9)
Hints for Selected Exercises and Problems 387(2)
Index of Symbols 389(4)
References 393(2)
Index 395
Christopher Heil is Professor of Mathematics at the Georgia Institute of Technology in Atlanta, Georgia. His research interests include harmonic analysis, time-frequency analysis, image processing, and more.