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Modern Analysis [Hardback]

(Michigan Tech University)
  • Format: Hardback, 592 pages, height x width: 235x156 mm, weight: 966 g, 565 equations, Contains 26 hardbacks
  • Series: Studies in Advanced Mathematics 26
  • Pub. Date: 20-Nov-1997
  • Publisher: CRC Press Inc
  • ISBN-10: 084937166X
  • ISBN-13: 9780849371660
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  • Price: 138,25 €*
  • * This title is out of print. Used copies may be available, but delivery only inside Baltic States
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Modern AnalysisLarger Image
  • Format: Hardback, 592 pages, height x width: 235x156 mm, weight: 966 g, 565 equations, Contains 26 hardbacks
  • Series: Studies in Advanced Mathematics 26
  • Pub. Date: 20-Nov-1997
  • Publisher: CRC Press Inc
  • ISBN-10: 084937166X
  • ISBN-13: 9780849371660
Other books in subject:
Permanent link: https://www.kriso.ee/db/9780849371660.html
Keywords:
A textbook for a graduate course in real and abstract analysis or a basic course on measure theory. Assumes knowledge of normed vector space, the basic theorems in linear algebra, basic set theory and standard notation. Includes the Bochner integral that is widely used by engineers and others but often difficult to find in the literature, information on introductory topology to help readers understand the Reisz representation theory and related topics, various proofs of partial differential equations not often found, and exercises that illustrate and extend the theorems. Annotation c. by Book News, Inc., Portland, Or.

Modern Analysis provides coverage of real and abstract analysis, offering a sensible introduction to functional analysis as well as a thorough discussion of measure theory, Lebesgue integration, and related topics. This significant study clearly and distinctively presents the teaching and research literature of graduate analysis:
  • Providing a fundamental, modern approach to measure theory
  • Investigating advanced material on the Bochner integral, geometric theory, and major theorems in Fourier Analysis Rn, including the theory of singular integrals and Milhin's theorem - material that does not appear in textbooks
  • Offering exceptionally concise and cardinal versions of all the main theorems about characteristic functions
  • Containing an original examination of sufficient statistics, based on the general theory of Radon measures
    With an ambitious scope, this resource unifies various topics into one volume succinctly and completely. The contents span basic measure theory in an abstract and concrete form, material on classic linear functional analysis, probability, and some major results used in the theory of partial differential equations. Two different proofs of the central limit theorem are examined as well as a straightforward approach to conditional probability and expectation.
    Modern Analysis provides ample and well-constructed exercises and examples. Introductory topology is included to help the reader understand such items as the Riesz theorem, detailing its proofs and statements. This work will help readers apply measure theory to probability theory, guiding them to understand the theorems rather than merely follow directions.
    • 1 Set Theory and General Topology
    		1(16)
    1 Set theory
    		1(3)
    2 General topology
    		4(5)
    3 Urysohn's lemma
    		9(3)
    4 Exercises
    		12(5)
    2 Compactness and Continuous Functions
    		17(16)
    1 Compactness in metric space
    		17(4)
    2 Compactness in spaces of continuous functions
    		21(3)
    3 Stone Weierstrass theorem
    		24(5)
    4 Exercises
    		29(4)
    3 Banach Spaces
    		33(16)
    1 Baire category theorem
    		33(3)
    2 Uniform boundedness closed graph and open mapping theorems
    		36(4)
    3 Hahn Banach theorem
    		40(5)
    4 Exercises
    		45(4)
    4 Hilbert Spaces
    		49(14)
    1 Finite dimensional normed linear space
    		53(2)
    2 Uniformly convex Banach spaces
    		55(5)
    3 Exercises
    		60(3)
    5 Calculus in Banach Space
    		63(24)
    1 The derivative
    		63(2)
    2 Finite dimensions
    		65(3)
    3 Higher order derivatives
    		68(2)
    4 Inverse function theorem
    		70(8)
    5 Ordinary differential equations
    		78(2)
    6 Exercises
    		80(7)
    6 Locally Convex Topological Vector Spaces
    		87(36)
    1 Separation theorems
    		91(6)
    2 The weak and weak* topologies
    		97(8)
    3 The Tychonoff fixed point theorem
    		105(5)
    4 Set-valued maps
    		110(5)
    5 Finite dimensional spaces
    		115(4)
    6 Exercises
    		119(4)
    7 Measures and Measurable Functions
    		123(12)
    1 XXX Algebras
    		123(1)
    2 Monotone classes and algebras
    		123(1)
    2 Monotone classes and algebras
    		123(9)
    3 Exercises
    		132(3)
    8 The Abstract Lebesgue Integral
    		135(14)
    1 The space L(1)
    		139(5)
    2 Double sums of nonnegative terms
    		144(1)
    3 Exercises
    		145(4)
    9 The Construction Of Measures
    		149(22)
    1 Outer measures
    		149(5)
    2 Regular measures
    		154(14)
    3 Lebesgue measure on R(1)
    		168(1)
    4 Exercises
    		168(3)
    10 Lebesgue Measure
    		171(18)
    1 Lebesgue measure
    		171(4)
    2 Iterated integrals
    		175(2)
    3 Change of variables
    		177(6)
    4 Polar coordinates
    		183(3)
    5 Exercises
    		186(3)
    11 Product Measure
    		189(14)
    1 Completion of product measure
    		195(4)
    2 Exercises
    		199(4)
    12 The L(p) Spaces
    		203(16)
    1 Basic inequalities and properties
    		203(4)
    2 Density of simple functions
    		207(2)
    3 Continuity of translation
    		209(1)
    4 Separability
    		210(1)
    5 Mollifiers and density of smooth functions
    		211(2)
    6 Exercises
    		213(6)
    13 Representation Theorems
    		219(26)
    1 Radon Nikodym Theorem
    		219(2)
    2 Vector measures
    		221(5)
    3 Representation theorems for the dual space of L(p) the XXX finite case
    		226(6)
    4 Riesz Representation theorem for non XXX finite measure spaces
    		232(5)
    5 The dual space of C(X)
    		237(4)
    6 Exercises
    		241(4)
    14 Fundamental Theorem of Calculus
    		245(18)
    1 The Vitali covering theorem
    		245(2)
    2 Differentiation with respect to Lebesgue measure
    		247(4)
    3 The change of variables theorem for multiple integrals
    		251(8)
    4 Exercises
    		259(4)
    15 General Radon Measures
    		263(20)
    1 Besicovitch covering theorem
    		263(5)
    2 Differentiation with respect to Radon measures
    		268(3)
    3 Slicing measures
    		271(6)
    4 Young measures
    		277(4)
    5 Exercises
    		281(2)
    16 Fourier Transforms
    		283(20)
    1 The Schwartz class
    		283(5)
    2 Fourier transforms of functions in L(2) (R(n))
    		288(5)
    3 Tempered distributions
    		293(6)
    4 Exercises
    		299(4)
    17 Probability
    		303(52)
    1 Random vectors
    		303(5)
    2 Conditional probability and independence
    		308(8)
    3 Conditional expectation
    		316(4)
    4 Conditional expectation given a XXX algebra
    		320(10)
    5 Strong law of large numbers
    		330(5)
    6 The normal distribution
    		335(3)
    7 The central limit theorem
    		338(4)
    8 The continuity theorem
    		342(6)
    9 Exercises
    		348(7)
    18 Weak Derivatives
    		355(16)
    1 Test functions and weak derivatives
    		355(4)
    2 Weak derivatives in L(p)(loc)
    		359(2)
    3 Morrey's inequality
    		361(2)
    4 Rademacher's theorem
    		363(3)
    5 Exercises
    		366(5)
    19 Hausdorff Measures
    		371(16)
    1 Steiner symmetrization
    		373(2)
    2 The isodiametric inequality
    		375(2)
    3 Hausdorff measures
    		377(1)
    4 Properties of Hausdorff measure
    		378(9)
    20 The Area Formula
    		387(32)
    1 Lipschitz mappings
    		387(8)
    2 The area formula for one to one Lipschitz mappings
    		395(3)
    3 Mappings that are not one to one
    		398(4)
    4 Surface measure
    		402(3)
    5 The divergence theorem
    		405(7)
    6 Exercises
    		412(7)
    21 The Coarea Formula
    		419(16)
    1 A determinant identity
    		420(1)
    2 The Coarea formula
    		421(11)
    3 Change of variables
    		432(1)
    4 Exercises
    		433(2)
    22 Fourier Analysis in R(n)
    		435(40)
    1 The Marcinkiewicz interpolation theorem
    		435(3)
    2 The Calderon Zygmund decomposition
    		438(2)
    3 Mihlin's theorem
    		440(13)
    4 Singular integrals
    		453(9)
    5 The Helmholtz decomposition of vector fields
    		462(7)
    6 Exercises
    		469(6)
    23 Integration for Vector Valued Functions
    		475(28)
    1 Strong and weak measurability
    		475(6)
    2 The Bochner integral
    		481(7)
    3 Measurable representatives
    		488(2)
    4 Vector measures
    		490(4)
    5 The Riesz representation theorem
    		494(5)
    6 Exercises
    		499(4)
    24 Convex Functions
    		503(26)
    1 Continuity properties of convex functions
    		503(3)
    2 Separation properties
    		506(3)
    3 Conjugate functions
    		509(2)
    4 Subgradients
    		511(8)
    5 Hilbert space
    		519(5)
    6 Exercises
    		524(5)
    Appendix 1: The Hausdorff Maximal theorem 529(6)
    1 Exercises 532(3)
    Appendix 2: Stone's Theorem and Partitions of Unity 535(10)
    1 General partitions of unity 540(1)
    2 A general metrization theorem 541(4)
    Appendix 3: Taylor Series and Analytic Functions 545(12)
    1 Taylor's formula 545(1)
    2 Analytic functions 546(6)
    3 Ordinary differential equations 552(5)
    Appendix 4: The Brouwer Fixed Point theorem 557(4)
    References 561(8)
    Index 569