This book is meant as a text for a first-year graduate course in analysis. In a sense, the subject matter covers the same topics as elementary calculus - linear algebra, differentiation, integration - but treated in a manner suitable for people who will be using it in further mathematical investigations. The book begins with point-set topology, essential for all analysis. The second part deals with the two basic spaces of analysis, Banach and Hilbert spaces. The book then turns to the subject of integration and measure. After a general introduction, it covers duality and representation theorems, some applications (such as Dirac sequences and Fourier transforms), integration and measures on locally compact spaces, the Riemann-Stjeltes integral, distributions, and integration on locally compact groups. Part four deals with differential calculus (with values in a Banach space). The next part deals with functional analysis. It includes several major spectral theorems of analysis, showing how one can extend to infinite dimensions certain results from finite-dimensional linear algebra; a discussion of compact and Fredholm operators; and spectral theorems for Hermitian operators. The final part, on global analysis, provides an introduction to differentiable manifolds. The text includes worked examples and numerous exercises, which should be viewed as an integral part of the book. The organization of the book avoids long chains of logical interdependence, so that chapters are as independent as possible. This allows a course using the book to omit material from some chapters without compromising the exposition of material from later chapters.
The second edition was published as Real Analysis, Addison- Wesley, 1983. The third edition has been reorganized. After a brief introduction to point set topology, some basic theorems on continuous functions, and the introduction of Banach and Hilbert spaces, integration theory is covered so it can fit in a one-semester course. Functional analysis follows for a second semester. Examples and exercises have been added, as well as material on integration pertaining to the real line. Functional analysis now includes the Gelfand transform, and basic spectral theorems for compact, bounded hermitian, and unbounded self-adjoint operators. Annotation c. Book News, Inc., Portland, OR (booknews.com)