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This book is based on lectures given at "Mekhmat", the Department of Mechanics and Mathematics at Moscow State University, one of the top mathematical departments worldwide, with a rich tradition of teaching functional analysis.

Featuring an advanced course on real and functional analysis, the book presents not only core material traditionally included in university courses of different levels, but also a survey of the most important results of a more subtle nature, which cannot be considered basic but which are useful for applications. Further, it includes several hundred exercises of varying difficulty with tips and references.

The book is intended for graduate and PhD students studying real and functional analysis as well as mathematicians and physicists whose research is related to functional analysis.

### Chapter 1. Metric and Topological Spaces

Abstract
In this chapter we consider the most important concepts of modern analysis — metric and topological spaces. Large parts of the chapter consist of definitions and examples (probably, already familiar to the reader from a course of calculus), moreover, some additional information about compact spaces needed for the sequel is given.
Vladimir I. Bogachev, Oleg G. Smolyanov

### Chapter 2. Fundamentals of Measure Theory

Abstract
In this chapter and the next two we present Lebesgue’s theory of measure and integral. In addition, we discuss connections between integration and differentiation.
Vladimir I. Bogachev, Oleg G. Smolyanov

### Chapter 3. The Lebesgue Integral

Abstract
In this chapter we study the Lebesgue integral and obtain fundamental results about passage to the limit under the integral sign and spaces of integrable functions. In addition, we prove very important theorems due to Radon and Nikodym and Fubini.
Vladimir I. Bogachev, Oleg G. Smolyanov

### Chapter 4. Connections between the Integral and Derivative

Abstract
In this chapter we briefly discuss integration by parts for the Lebesgue integral and some related differentiation problems.
Vladimir I. Bogachev, Oleg G. Smolyanov

### Chapter 5. Normed and Euclidean Spaces

Abstract
In this chapter we discuss some basic geometric and topological properties of normed and Euclidean spaces, which are the most important types of spaces of functional analysis.
Vladimir I. Bogachev, Oleg G. Smolyanov

### Chapter 6. Linear Operators and Functionals

Abstract
In this chapter we discuss one of the central concepts of functional analysis — linear operators. We first establish the three most important results about general linear operators: the Banach–Steinhaus theorem, Banach’s inverse mapping theorem, and the closed graph theorem.
Vladimir I. Bogachev, Oleg G. Smolyanov

### Chapter 7. Spectral Theory

Abstract
This chapter is devoted to a branch of the theory of operators very important for applications — spectral theory. More than any other chapter of the present book spectral theory owes its creation and intensive development to problems in natural sciences, in particular, in mechanics, physics, and chemistry.
Vladimir I. Bogachev, Oleg G. Smolyanov

### Chapter 8. Locally Convex Spaces and Distributions

Abstract
In this chapter we study linear spaces more general than normed spaces. For many problems in applications the framework of normed spaces turns out to be too stringent; a typical example is the space of infinitely differentiable functions that possesses its natural convergence, which, however, cannot be described by means of a norm. Some elements of this theory have already been encountered in our discussion of weak topologies.
Vladimir I. Bogachev, Oleg G. Smolyanov

### Chapter 9. The Fourier Transform and Sobolev Spaces

Abstract
In this chapter we discuss one of the most classical objects of analysis — the Fourier transform, and also give an elementary introduction to the theory of Sobolev spaces, which plays a very important role in modern analysis. It will not be an exaggeration to say that Sobolev spaces are applied everywhere where the derivative and integral are used. Along with Lebesgue’s integral and Banach spaces, Sobolev spaces belong to the greatest achievements of analysis in the XX century, determining its modern appearance. Of course, this introduction cannot pretend even to be a brief course of the theory of Sobolev spaces.
Vladimir I. Bogachev, Oleg G. Smolyanov

### Chapter 10. Unbounded Operators and Operator Semigroups

Abstract
This chapter is devoted to fundamentals of the spectral theory of unbounded selfadjoint operators and some elements of the theory of operator semigroups. Some of the principal applications of these theories are connected with partial differential equations and mathematical physics, in particular, quantum mechanics, but there are also important applications in many other areas of mathematics, for example, in the theory of random processes and geometry.
Vladimir I. Bogachev, Oleg G. Smolyanov

### Chapter 11. Banach Algebras

Abstract
In this chapter we give a brief introduction to the theory of Banach algebras. As well as operator theory, this theory has important applications in physics. One of the most important Banach algebras is the algebra of operators. Considerations of this chapter shed a new light on some already encountered objects and also complement some results obtained above.
Vladimir I. Bogachev, Oleg G. Smolyanov

### Chapter 12. Infinite-Dimensional Analysis

Abstract
In this chapter we discuss foundations of differential calculus in infinite–dimensional spaces and some related questions. In the finite–dimensional case there are two different types of differentiability: differentiability at a point based on the consideration of increments of the function and also a global differentiability based on the consideration of the derivative as some independent object (as this is done in the theory of distributions and in the theory of Sobolev spaces).
Vladimir I. Bogachev, Oleg G. Smolyanov