This book consists of three major parts. The first two parts deal with general mathematical concepts and certain areas of operator theory. The third part is devoted to ill-posed problems. It can be read independently of the first two parts and presents a good example of applying the methods of calculus and functional analysis. The first part "Basic Concepts" briefly introduces the language of set theory and concepts of abstract, linear and multilinear algebra. Also introduced are the language of topology and fundamental concepts of calculus: the limit, the differential, and the integral. A special section is devoted to analysis on manifolds. The second part "Operators" describes the most important function spaces and operator classes for both linear and nonlinear operators. Different kinds of generalized functions and their transformations are considered. Elements of the theory of linear operators are presented. Spectral theory is given a special focus. The third part "Ill-Posed Problems" is devoted to problems of mathematical physics, integral and operator equations, evolution equations and problems of integral geometry. It also deals with problems of analytic continuation. Detailed coverage of the subjects and numerous examples and exercises make it possible to use the book as a textbook on some areas of calculus and functional analysis. It can also be used as a reference textbook because of the extensive scope and detailed references with comments.

Table of Contents:
Basic concepts Chapter 1. Set theory 1.1. Sets 1.2. Correspondences 1.3. Relations 1.4. Induction 1.5. Natural numbers Chapter 2. Algebra 2.1. Abstract algebra 2.2. Linear algebra 2.3. Multilinear algebra Chapter 3. Calculus 3.1. Limit 3.2. Differential 3.3. Integral 3.4. Analysis on manifolds Operators Chapter 4. Linear operators 4.1. Hilbert spaces 4.2. Fourier series 4.3. Function spaces 4.4. Fourier transform 4.5. Bounded linear operators 4.6. Compact linear operators 4.7. Self-adjoint operators 4.8. Spectra of operators 4.9. Spectral theorem 4.10. Operator exponential Chapter 5. Nonlinear operators 5.1. Fixed points 5.2. Saddle points 5.3. Monotonic operators 5.4. Nonlinear contractions 5.5. Degree theory Ill-posed problems Chapter 6. Classic problems 6.1. Mathematical description of the laws of physics 6.2. Equations of the first order 6.3. Classification of differential equations of the second order 6.4. Elliptic equations 6.5. Hyperbolic and parabolic equations 6.6. The notion of well-posedness Chapter 7. Ill-posed problems 7.1. Ill-posed Cauchy problems 7.2. Analytic continuation and interior problems 7.3. Weakly and strongly ill-posed problems. Problems of differentiation 7.4. Reducing ill-posed problems to integral equations Chapter 8. Physical problems leading to ill-posed problems 8.1. Interpretation of measurement data from physical devices 8.2. Interpretation of gravimetric data 8.3. Problems for the diffusion equation 8.4. Determining physical fields from the measurements data 8.5. Tomography Chapter 9. Operator and integral equations 9.1. Definitions of well-posedness 9.2. Regularization 9.3. Linear operator equations 9.4. Integral equations with weak singularities 9.5. Scalar Volterra equations 9.6. Volterra operator equations Chapter 10. Evolution equations 10.1. Cauchy problem and semigroups of operators 10.2. Equations in a Hilbert space 10.3. Equations with variable operator 10.4. Equations of the second order 10.5. Well-posed and ill-posed Cauchy problems 10.6. Equations with integro-differential operators Chapter 11. Problems of integral geometry 11.1. Statement of problems of integral geometry 11.2. The Radon problem 11.3. Reconstructing a function from spherical means 11.4. Planar problem of the general form 11.5. Spatial problems of the general form 11.6. Problems of the Volterra type for manifolds invariant with respect to the translation group 11.7. Planar problems of integral geometry with a perturbation Chapter 12. Inverse problems 12.1. Statement of inverse problems 12.2. Inverse dynamic problem. A linearization method 12.3. A general method for studying inverse problems for hyperbolic equations 12.4. The connection between inverse problems for hyperbolic, elliptic, and parabolic equations 12.5. Problems of determining a Riemannian metric Chapter 13. Several areas of the theory of ill-posed problems, inverse problems, and applications Bibliography Index