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E-grāmata: Course in Abstract Analysis

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  • Formāts: 367 pages
  • Sērija : Graduate Studies in Mathematics
  • Izdošanas datums: 10-Mar-2012
  • Izdevniecība: American Mathematical Society
  • ISBN-13: 9780821891605
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Course in Abstract AnalysisPalielināt
  • Formāts: 367 pages
  • Sērija : Graduate Studies in Mathematics
  • Izdošanas datums: 10-Mar-2012
  • Izdevniecība: American Mathematical Society
  • ISBN-13: 9780821891605
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This book covers topics appropriate for a first-year graduate course preparing students for the doctorate degree. The first half of the book presents the core of measure theory, including an introduction to the Fourier transform. This material can easily be covered in a semester. The second half of the book treats basic functional analysis and can also be covered in a semester. After the basics, it discusses linear transformations, duality, the elements of Banach algebras, and C*-algebras. It concludes with a characterization of the unitary equivalence classes of normal operators on a Hilbert space. The book is self-contained and only relies on a background in functions of a single variable and the elements of metric spaces. Following the author's belief that the best way to learn is to start with the particular and proceed to the more general, it contains numerous examples and exercises.
Preface xi
Chapter 1 Setting the Stage
1(40)
§1.1 Riemann-Stieltjes integrals
1(11)
§1.2 Metric spaces redux
12(9)
§1.3 Normed spaces
21(8)
§1.4 Locally compact spaces
29(8)
§1.5 Linear functionals
37(4)
Chapter 2 Elements of Measure Theory
41(52)
§2.1 Positive linear functionals on C(X)
41(1)
§2.2 The Radon measure space
42(9)
§2.3 Measurable functions
51(5)
§2.4 Integration with respect to a measure
56(15)
§2.5 Convergence theorems
71(7)
§2.6 Signed measures
78(6)
§2.7 Lp-spaces
84(9)
Chapter 3 A Hilbert Space Interlude
93(14)
§3.1 Introduction to Hilbert space
93(5)
§3.2 Orthogonality
98(5)
§3.3 The Riesz Representation Theorem
103(4)
Chapter 4 A Return to Measure Theory
107(64)
§4.1 The Lebesgue-Radon-Nikodym Theorem
107(7)
§4.2 Complex functions and measures
114(5)
§4.3 Linear functionals on C(X)
119(5)
§4.4 Linear functionals on Co(X)
124(3)
§4.5 Functions of bounded variation
127(2)
§4.6 Linear functionals on Lp-spaces
129(4)
§4.7 Product measures
133(8)
§4.8 Lebesgue measure on Rd
141(3)
§4.9 Differentiation on Rd
144(7)
§4.10 Absolutely continuous functions
151(5)
§4.11 Convolution
156(5)
§4.12 The Fourier transform
161(10)
Chapter 5 Linear Transformations
171(42)
§5.1 Basics
171(4)
§5.2 Orthonormal basis
175(4)
§5.3 Isomorphic Hilbert spaces
179(4)
§5.4 The adjoint
183(7)
§5.5 The direct sum of Hilbert spaces
190(5)
§5.6 Compact linear transformations
195(7)
§5.7 The Spectral Theorem
202(3)
§5.8 Some applications of the Spectral Theorem
205(5)
§5.9 Unitary equivalence
210(3)
Chapter 6 Banach Spaces
213(24)
§6.1 Finite-dimensional spaces
213(3)
§6.2 Sums and quotients of normed spaces
216(4)
§6.3 The Hahn-Banach Theorem
220(6)
§6.4 Banach limits
226(2)
§6.5 The Open Mapping and Closed Graph Theorems
228(4)
§6.6 Complemented subspaces
232(2)
§6.7 The Principle of Uniform Boundedness
234(3)
Chapter 7 Locally Convex Spaces
237(14)
§7.1 Basics of locally convex spaces
237(6)
§7.2 Metrizable locally convex spaces
243(1)
§7.3 Geometric consequences
244(7)
Chapter 8 Duality
251(26)
§8.1 Basics of duality
251(9)
§8.2 The dual of a quotient space and of a subspace
260(3)
§8.3 Reflexive spaces
263(3)
§8.4 The Krein-Milman Theorem
266(4)
§8.5 The Stone-Weierstrass Theorem
270(7)
Chapter 9 Operators on a Banach Space
277(10)
§9.1 The adjoint
277(5)
§9.2 Compact operators
282(5)
Chapter 10 Banach Algebras and Spectral Theory
287(32)
§10.1 Elementary properties and examples
287(3)
§10.2 Ideals and quotients
290(2)
§10.3 Analytic functions
292(5)
§10.4 The spectrum
297(5)
§10.5 The spectrum of an operator
302(8)
§10.6 The spectrum of a compact operator
310(4)
§10.7 Abelian Banach algebras
314(5)
Chapter 11 C*-Algebras
319(36)
§11.1 Elementary properties and examples
319(4)
§11.2 Abelian C*-algebras
323(4)
§11.3 Positive elements in a C*-algebra
327(5)
§11.4 A functional calculus for normal operators
332(10)
§11.5 The commutant of a normal operator
342(3)
§11.6 Multiplicity theory
345(10)
Appendix
355(4)
§A.1 Baire Category Theorem
355(1)
§A.2 Nets
356(3)
Bibliography 359(2)
List of Symbols 361(2)
Index 363
John B. Conway, George Washington University, Washington, DC, USA
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