Real and Functional Analysis (Graduate Texts in Mathematics, 142)

Real and Functional Analysis is essentially a third edition of Real Analysis, and is designed for a basic graduate course. The book 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 systematically so it can fit in a one-semester course. The functional analysis then follows, for a second semester. A number of examples and exercises have been added, as well as some material about integration pertaining specifically to the real line (e.g., Dirac sequence approximation and Fourier analysis in connection with functions of bounded variation and Stieltjes integrals). The functional analysis has been rounded off to include the Gelfand transform on C*-algebras as well as the basic spectral theorems, for compact operators, bounded hermitian operators, and unbounded self-adjoint operators.A number of topics are included for complementary reading, for instance the law of large numbers and Stokes' theorem on manifolds (even with singularities). The inclusion of such additional material also makes the book useful as a reference source.