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Real Analysis
Rami Shakarchi, Elias M. Stein
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Naturwissenschaften, Medizin, Informatik, Technik / Analysis
Beschreibung
Real Analysis is the third volume in the Princeton Lectures in Analysis, a series of four textbooks that aim to present, in an integrated manner, the core areas of analysis. Here the focus is on the development of measure and integration theory, differentiation and integration, Hilbert spaces, and Hausdorff measure and fractals. This book reflects the objective of the series as a whole: to make plain the organic unity that exists between the various parts of the subject, and to illustrate the wide applicability of ideas of analysis to other fields of mathematics and science.
After setting forth the basic facts of measure theory, Lebesgue integration, and differentiation on Euclidian spaces, the authors move to the elements of Hilbert space, via the L2 theory. They next present basic illustrations of these concepts from Fourier analysis, partial differential equations, and complex analysis. The final part of the book introduces the reader to the fascinating subject of fractional-dimensional sets, including Hausdorff measure, self-replicating sets, space-filling curves, and Besicovitch sets. Each chapter has a series of exercises, from the relatively easy to the more complex, that are tied directly to the text. A substantial number of hints encourage the reader to take on even the more challenging exercises.
As with the other volumes in the series, Real Analysis is accessible to students interested in such diverse disciplines as mathematics, physics, engineering, and finance, at both the undergraduate and graduate levels.
Also available, the first two volumes in the Princeton Lectures in Analysis:
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Hilbert space, Holomorphic function, Summation, Fubini's theorem, Lebesgue integration, Lecture, Linear map, Dimension, Fourier series, Two-dimensional space, Cauchy–Schwarz inequality, Addition, Complex analysis, Infimum and supremum, Interval (mathematics), Big O notation, Uniform convergence, Suggestion, Unit interval, Bernhard Riemann, Variable (mathematics), Bounded function, Union (set theory), Compact space, Lebesgue measure, Support (mathematics), Special case, Dimension (vector space), Disjoint union, Fourier transform, Continuous function, Cauchy sequence, Rectangle, Simple function, Continuous function (set theory), Theorem, Bounded set (topological vector space), Boundary value problem, Quantity, Existential quantification, Subsequence, Monotonic function, Hausdorff measure, Riemann integral, Cantor set, Poisson kernel, Exterior (topology), Mathematical induction, Borel set, Diameter, Projection (linear algebra), Subset, Step function, Measure (mathematics), Sign (mathematics), Linear combination, Mathematics, Orthonormal basis, Measurable function, Parseval's identity, Pointwise, Scientific notation, Equivalence class, Bounded variation, Axiom of choice, Absolute continuity, Corollary, Differentiable function, Monotone convergence theorem, Characteristic function (probability theory)