This new edition of the widely used analysis book continues tocover real analysis in greater detail and at a more advanced levelthan most books on the subject. Encompassing several subjects thatunderlie much of modern analysis, the book focuses on measure andintegration theory, point set topology, and the basics offunctional analysis. It illustrates the use of the general theoriesand introduces readers to other branches of analysis such asFourier analysis, distribution theory, and probabilitytheory.
This edition is bolstered in content as well as in scope-extendingits usefulness to students outside of pure analysis as well asthose interested in dynamical systems. The numerous exercises,extensive bibliography, and review chapter on sets and metricspaces make Real Analysis: Modern Techniques and TheirApplications, Second Edition invaluable for students ingraduate-level analysis courses. New features include:
* Revised material on the n-dimensional Lebesgue integral.
* An improved proof of Tychonoff's theorem.
* Expanded material on Fourier analysis.
* A newly written chapter devoted to distributions and differentialequations.
* Updated material on Hausdorff dimension and fractal dimension.
This book develops the abstract theory along with a well-chosen selection of concrete examples that exemplify the results and show the breadth of their applicability. After a preliminary chapter containing the necessary background material on Banach algebras and spectral theory, the text sets out the general theory of locally compact groups and their unitary representations, followed by a development of the more specific theory of analysis on Abelian groups and compact groups. There is an extensive chapter on the theory of induced representations and its applications, and the book concludes with a more informal exposition on the theory of representations of non-Abelian, non-compact groups.
Featuring extensive updates and new examples, the Second Edition:Adds a short section on von Neumann algebras Includes Mark Kac’s simple proof of a restricted form of Wiener’s theorem Explains the relation between SU(2) and SO(3) in terms of quaternions, an elegant method that brings SO(4) into the picture with little effort Discusses representations of the discrete Heisenberg group and its central quotients, illustrating the Mackey machine for regular semi-direct products and the pathological phenomena for nonregular ones
A Course in Abstract Harmonic Analysis, Second Edition serves as an entrée to advanced mathematics, presenting the essentials of harmonic analysis on locally compact groups in a concise and accessible form.