Functional Analysis

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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 applications 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 Lebesque integral and study of the spaces L2 and Lp 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 completelycontinuous 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 extension so 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, Profs. 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. Table of Contents Differentiation Lebesgue’s theorem on the derivative of a monotonic function Some immediate consequences of Lebesque’s theorem Interval functions The Lebesque integral Definition and fundamental properties Indefinite integrals. Absolutely continuous functions The space L2 and its linear functionals. Lp spaces Functions of several variables Other definitions of the Lebesgue integral The stieltjes integral and its generalizations Linear functionals on the space of continuous functions Generalization of the Stieltjes integral The Daniel integral Integral equation. Linear transformations Integral equations The method of successive approximations The Fredholm alternative Fredholm determinants Another method, based on complete continuity Applications to potential theory Hilbert and banach spaces Hilbert space Banach spaces Completely continuous symmetric transformations of Hilbert space Existence of characteristic elements. Theorem on series development Transformation with symmetric kernel Applications for the vibrating- string problem and to almost periodic functions Bounded symmetric, unitary and normal transformations of Hilbert Space Unitary and normal transformations Unitary transformations of the space L2 Unbounded linear transformations of Hilbert space Self-adjoint transformations, functional calculus, spectrum, perturbations, Groups and semi groups of transformations Spectral theories for linear transformations of general type.