This noted text, highly regarded in the field, discusses modern theories of differentiation and integration and the principal problems and methods of handling integral equations and linear functionals and transformations.
Part One begins with a direct proof of the Lebesgue theorem on the differentiation of monotonic functions and its application to the study of the relations between the derivatives and the integrals of interval functions. This is followed by construction of the theory of the Lebesgue integral and study of the spaces L2
and their linear functionals. The Stieltjes integral and its generalizations are introduced in terms of linear operations on the space of continuous functions.
Part Two begins with a chapter on integral equations. The authors then present several methods for arriving at the Fredholm alternative, which are subsequently applied to completely continuous functional equations of general type on either a Hilbert space or a Banach space. Symmetric completely continuous linear transformations are studied separately.
Next comes development of the spectral theory of self-adjoint transformations, either bounded or unbounded, of Hilbert space. Also considered are the problem of the extensions of unbounded symmetric transformations, functions of a self-adjoint transformation and the study of the spectrum and its perturbations, Stone's theorem on groups of unitary transformations, certain ergodic theorems, and more.
Finally, the authors survey the beginnings of the spectral theory of linear transformations, including applications of methods from the theory of functions and Von Neumann's theory of spectral sets.
Throughout the text, Professors Riesz and Sz.-Nagy have sought to present the principal problems and the methods for handling them, rather than attempting to study in detail all possible generalizations. The result is a classic, highly useful exposition that will be of interest to advanced undergraduates and graduate students of mathematics and related fields.
Reprint of the Frederick Ungar Publishing Co., New York, 1955 edition.