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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.
Real Analysis: Measure Theory, Integration, and Hilbert Spaces
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.
Real Analysis, Measure Theory, Integration, And Hilbert Spaces
The final three chapters deal with other examples of Hilbert spaces, abstract measure and integration, and the Hausdorff measure. As one might expect, several of the main results of the first volume are re-examined in this one.
argo-karaganda.kz/scripts/jigetehig/2757.php This, of course, is one of the features of the authors' approach, which makes it easier to motivate the more abstract material in this book and also highlights the connections between different branches of analysis. The fourth volume, which is to cover functional analysis, distributions, and parts of probability theory, will complete the series.
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