The book is based on the course of lectures on calculus and functional analysis and several special courses given by the authors at Novosibirsk State University. It also includes results of research carried out at the Institute of Mathematics of the Siberian Branch of the Russian Academy of Sciences. A brief introduction to the language of set theory and elements of abstract, linear, and multilinear algebra is provided. The language of topology is introduced and fundamental concepts of analysis for vector spaces and manifolds are described in detail. The most often used spaces of smooth and generalized functions, their transformations, and the classes of linear and nonlinear operators are considered. Special attention is given to spectral theory and the fixed point theorems. A brief presentation of degree theory is provided. The part devoted to ill-posed problems includes a description of partial differential equations, integral and operator equations, and problems of integral geometry. The book can serve as a textbook or reference on functional analysis. It contains many examples. It can also be of interest to specialists in the above fields.

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