S. V. Fomin – författare

Self-contained and comprehensive, this elementary introduction to real and functional analysis is readily accessible to those with background in advanced calculus. It covers basic concepts and introductory principles in set theory, metric spaces, topological and linear spaces, linear functionals and linear operators, and much more. 350 problems. 1970 edition.

First six chapters include theory of fields and sufficient conditions for weak and strong extrema. Chapter seven considers application of variation methods to systems with infinite degrees of freedom, and Chapter eight deals with direct methods in the calculus of variations. Problems follow each chapter and the 2 appendices.

I. Measures and quasimeasures. Integration.- 1. Realvalued measures on algebras of sets.- 1.1. Premeasures.- 1.2. Same tests for ?-additivity of premeasures.- 1.3. Measurable and topological Radon spaces.- 1.4. Cylindrical measures.- 2. Cylinder sets and cylindrical functions.- 2.1. General definition of cylinder set.- 2.2. Cylinder sets in a linear space X.- 2.3. Measurable linear space.- 2.4. Cylindrical functions.- 3. Quasimeasures. Integration.- 3.1. Quasimeasures.- 3.2. Integral with respect to a quasimeasure.- 3.3. Quasimeasures in a measurable linear space.- 3.4. Positive quasimeasures.- 3.5. Integration of noncylindrical functions.- 4. Supplement: Some notions related to the topology of linear spaces.- 4.1. Prenorms.- 4.2. Locally convex spaces.- 4.3. Duality of linear spaces.- 4.4. Rigged Hilbert spaces.- 4.5. Polars.- 4.6. Nuclear topology.- 4.7. Compactness.- 5. Chapter I: Supplementary remarks and historical comments.- II. Gaussian measures in Hilbert space.- 1. Gaussian measures in finite-dimensional spaces.- 1.1. Characteristic functional and density.- 1.2. Computation of certain integrals.- 1.3. Integration by parts.- 1.4. Solution of the Cauchy problem.- 2. Gaussian measures in Hilbert space.- 2.1. ?-additivity for a Gaussian cylindrical measure.- 2.2. Some transformations of Gaussian measures in X.- 2.3. Computation of integrals.- 2.4. Gaussian cylindrical measures with arbitrary correlation operator.- 3. Measurable linear functionals and operators.- 3.1. Measurable linear functionals.- 3.2. Measurable linear operators.- 3.3. Integration by parts.- 3.4. Expansion into orthogonal polynomials.- 4. Absolute continuity of Gaussian measures.- 4.1. Equivalence of measures in a product space.- 4.2. Equivalence of Gaussian measures which differ by their means.- 4.3. Equivalence of Gaussian measures with distinct correlation operators.- 4.4. Absolute continuity of measures obtained from Gaussian measures by certain transformations of space.- 5. Fourier-Wiener transformation.- 5.1. Fourier transformation with respect to a Gaussian measure.- 5.2. Fourier-Wiener transformation of entire nmctions.- 5.3. Connection between the Fourier-Wiener transformation and orthogonal polynomials.- 6. Complexvalued Gaussian quasimeasures.- 6.1. Feynman integrals.- 6.2. Integration of analytic functionals.- 6.3. Computation of certain Feynman integrals.- 7. Chapter II: Supplementary re marks and historical comments.- III. Measures in linear topological spaces.- 1. ?-additivity conditions for nonnegative cylindrical measures in the space X' dual to a locally convex space X.- 1.1. Sufficient conditions for ?-additivity. Strong regularity.- 1.2. Necessary conditions for ?-additivityM.- 1.3. The Hilbert space case.- 1.4. Integral representations of the group of unitary operators.- 1.5. Continuous cylindrical measures.- 2. Sequences of Radon measures.- 2.1. Weak compaetness in a spaee of measures.- 2.2. Weak completeness of spaees of measures.- 2.3. Properties of R-spaces.- 2.4. Examples of R-spaces.- 2.5. Weak compaetness of a family of measures in a space X'.- 3. Chapter III: Supplementary remarks and historical comments.- IV. Differentiable measures and distributions.- 1. Differentiable functions, differentiable expressions.- 1.1. Derivatives of a vector function.- 1.2. Higher order derivatives.- 1.3. Linear differential expressions.- 1.4. Symmetrie and dissipative differential operators.- 2. Differentiable measures.- 2.1. Derivative of a measure.- 2.2. The logarithmie derivative.- 2.3. The derivative of a measure as an element of the dual space.- 2.4. Higher order derivatives.- 3. Distributions and generalized functions.- 3.1. Test functions and measures.- 3.2. Distributions. Operations on distributions.- 3.3. Generalized funetions and kernels.- 3.4. Fourier transformation of distributions.- 3.5. Differential expressions for distributions.- 4. Positive definiteness. Quasi-invariant distributions and bidistributions.- 4.1. Positive distri

Ergodic theory is one of the few branches of mathematics which has changed radically during the last two decades. Before this period, with a small number of exceptions, ergodic theory dealt primarily with averaging problems and general qualitative questions, while now it is a powerful amalgam of methods used for the analysis of statistical properties of dyna­ mical systems. For this reason, the problems of ergodic theory now interest not only the mathematician, but also the research worker in physics, biology, chemistry, etc. The outline of this book became clear to us nearly ten years ago but, for various reasons, its writing demanded a long period of time. The main principle, which we adhered to from the beginning, was to develop the approaches and methods or ergodic theory in the study of numerous concrete examples. Because of this, Part I of the book contains the description of various classes of dynamical systems, and their elementary analysis on the basis of the fundamental notions of ergodicity, mixing, and spectra of dynamical systems. Here, as in many other cases, the adjective" elementary" i~ not synonymous with "simple. " Part II is devoted to "abstract ergodic theory. " It includes the construc­ tion of direct and skew products of dynamical systems, the Rohlin-Halmos lemma, and the theory of special representations of dynamical systems with continuous time. A considerable part deals with entropy.

Ergodic theory is one of the few branches of mathematics which has changed radically during the last two decades. Before this period, with a small number of exceptions, ergodic theory dealt primarily with averaging problems and general qualitative questions, while now it is a powerful amalgam of methods used for the analysis of statistical properties of dyna­ mical systems. For this reason, the problems of ergodic theory now interest not only the mathematician, but also the research worker in physics, biology, chemistry, etc. The outline of this book became clear to us nearly ten years ago but, for various reasons, its writing demanded a long period of time. The main principle, which we adhered to from the beginning, was to develop the approaches and methods or ergodic theory in the study of numerous concrete examples. Because of this, Part I of the book contains the description of various classes of dynamical systems, and their elementary analysis on the basis of the fundamental notions of ergodicity, mixing, and spectra of dynamical systems. Here, as in many other cases, the adjective" elementary" i~ not synonymous with "simple. " Part II is devoted to "abstract ergodic theory. " It includes the construc­ tion of direct and skew products of dynamical systems, the Rohlin-Halmos lemma, and the theory of special representations of dynamical systems with continuous time. A considerable part deals with entropy.