Basic Real Analysis [electronic resource] : Along with a companion volume Advanced Real Analysis / by Anthony W. Knapp.

By: Knapp, Anthony W [author.]
Contributor(s): SpringerLink (Online service)
Material type: TextTextSeries: Cornerstones: Publisher: Boston, MA : Birkhäuser Boston, 2005Description: XXIV, 656 p. online resourceContent type: text Media type: computer Carrier type: online resourceISBN: 9780817644413Subject(s): Mathematics | Mathematical analysis | Analysis (Mathematics) | Fourier analysis | Measure theory | Differential equations | Functions of real variables | Topology | Mathematics | Analysis | Measure and Integration | Real Functions | Fourier Analysis | Topology | Ordinary Differential EquationsAdditional physical formats: Printed edition:: No titleDDC classification: 515 LOC classification: QA299.6-433Online resources: Click here to access online
Contents:
Theory of Calculus in One Real Variable -- Metric Spaces -- Theory of Calculus in Several Real Variables -- Theory of Ordinary Differential Equations and Systems -- Lebesgue Measure and Abstract Measure Theory -- Measure Theory for Euclidean Space -- Differentiation of Lebesgue Integrals on the Line -- Fourier Transform in Euclidean Space -- Lp Spaces -- Topological Spaces -- Integration on Locally Compact Spaces -- Hilbert and Banach Spaces.
In: Springer eBooksSummary: Basic Real Analysis and Advanced Real Analysis (available separately or together as a Set) systematically develop those concepts and tools in real analysis that are vital to every mathematician, whether pure or applied, aspiring or established. These works present a comprehensive treatment with a global view of the subject, emphasizing the connections between real analysis and other branches of mathematics. Key topics and features of Basic Real Analysis: * Early chapters treat the fundamentals of real variables, sequences and series of functions, the theory of Fourier series for the Riemann integral, metric spaces, and the theoretical underpinnings of multivariable calculus and differential equations * Subsequent chapters develop the Lebesgue theory in Euclidean and abstract spaces, Fourier series and the Fourier transform for the Lebesgue integral, point-set topology, measure theory in locally compact Hausdorff spaces, and the basics of Hilbert and Banach spaces * The subjects of Fourier series and harmonic functions are used as recurring motivation for a number of theoretical developments * The development proceeds from the particular to the general, often introducing examples well before a theory that incorporates them * The text includes many examples and hundreds of problems, and a separate 55-page section gives hints or complete solutions for most of the problems Basic Real Analysis requires of the reader only familiarity with some linear algebra and real variable theory, the very beginning of group theory, and an acquaintance with proofs. It is suitable as a text in an advanced undergraduate course in real variable theory and in most basic graduate courses in Lebesgue integration and related topics. Because it focuses on what every young mathematician needs to know about real analysis, the book is ideal both as a course text and for self-study, especially for graduate students preparing for qualifying examinations. Its scope and approach will appeal to instructors and professors in nearly all areas of pure mathematics, as well as applied mathematicians working in analytic areas such as statistics, mathematical physics, and differential equations. Indeed, the clarity and breadth of Basic Real Analysis make it a welcome addition to the personal library of every mathematician.
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Theory of Calculus in One Real Variable -- Metric Spaces -- Theory of Calculus in Several Real Variables -- Theory of Ordinary Differential Equations and Systems -- Lebesgue Measure and Abstract Measure Theory -- Measure Theory for Euclidean Space -- Differentiation of Lebesgue Integrals on the Line -- Fourier Transform in Euclidean Space -- Lp Spaces -- Topological Spaces -- Integration on Locally Compact Spaces -- Hilbert and Banach Spaces.

Basic Real Analysis and Advanced Real Analysis (available separately or together as a Set) systematically develop those concepts and tools in real analysis that are vital to every mathematician, whether pure or applied, aspiring or established. These works present a comprehensive treatment with a global view of the subject, emphasizing the connections between real analysis and other branches of mathematics. Key topics and features of Basic Real Analysis: * Early chapters treat the fundamentals of real variables, sequences and series of functions, the theory of Fourier series for the Riemann integral, metric spaces, and the theoretical underpinnings of multivariable calculus and differential equations * Subsequent chapters develop the Lebesgue theory in Euclidean and abstract spaces, Fourier series and the Fourier transform for the Lebesgue integral, point-set topology, measure theory in locally compact Hausdorff spaces, and the basics of Hilbert and Banach spaces * The subjects of Fourier series and harmonic functions are used as recurring motivation for a number of theoretical developments * The development proceeds from the particular to the general, often introducing examples well before a theory that incorporates them * The text includes many examples and hundreds of problems, and a separate 55-page section gives hints or complete solutions for most of the problems Basic Real Analysis requires of the reader only familiarity with some linear algebra and real variable theory, the very beginning of group theory, and an acquaintance with proofs. It is suitable as a text in an advanced undergraduate course in real variable theory and in most basic graduate courses in Lebesgue integration and related topics. Because it focuses on what every young mathematician needs to know about real analysis, the book is ideal both as a course text and for self-study, especially for graduate students preparing for qualifying examinations. Its scope and approach will appeal to instructors and professors in nearly all areas of pure mathematics, as well as applied mathematicians working in analytic areas such as statistics, mathematical physics, and differential equations. Indeed, the clarity and breadth of Basic Real Analysis make it a welcome addition to the personal library of every mathematician.

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