Functional Analysis
Vol. I
Page
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Book
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Chapter
Measurable Functions
The concept of a measurable function was introduced by Lebesgue when constructing integration theory. Later, Luzin established the so-called C-property of measurable functions which, roughly speaking, can be form...
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Chapter
Measures in the Products of Spaces. Fubini Theorem
In this chapter, we show how to equip with a measure a Cartesian product of two or several measure spaces. The central result of this chapter is the Fubini theorem on the reduction of double integrals to itera...
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Chapter
Linear Continuous Functionals and Dual Spaces
In this chapter, we continue our study of Banach and Hilbert spaces. Here, we mainly consider linear functionals, i.e., additive, homogeneous, and continuous number functions given on such spaces. The problems...
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Chapter
Compact Operators. Equations with Compact Operators
In this chapter, we study an important class of linear continuous operators, namely, compact (or completely continuous) operators. On the one hand, compact operators are interesting because they inherit many p...
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Chapter
Elements of the Theory of Generalized Functions
In this chapter, we present the principles of the theory of generalized functions of the Sobolev-Schwartz-type. We consider classical spaces of test functions D(ℝ N ) and S(ℝ ...
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Chapter
Linear Normed Spaces and Hilbert Spaces
This chapter, in fact, starts our study of functional analysis. Here, we consider linear normed spaces, i.e., sets equipped with the structure of a linear space and a norm (an analog of the notion of the lengt...
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Chapter
Linear Continuous Operators
Linear continuous operators constitute one of the most important and well studied classes of map**s of linear normed spaces. In this chapter, we present the elementary theory of such operators. Linear contin...
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Chapter
Spectral Decomposition of Compact Selfadjoint Operators. Analytic Functions of Operators
The well-known theorem from linear algebra on the reducibility of an Hermitian matrix to the diagonal form can be reformulated as follows: For any selfadjoint operator A acting on the finite-dimensional Hilbert s...
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Chapter
Measure Theory
The general concept of a measure of a set is an abstract analogue of notions such as length, area, volume, mass, charge, etc. Having appeared at the beginning of this century in the papers by Lebesgue devoted ...
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Chapter
Theory of Integration
The concept of the Riemann integral is well known in mathematical analysis. Its essential shortage is connected with the fact that functions integrable according to Riemann must have “not too many” discontinui...
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Chapter
Absolute Continuity and Singularity of Measures, Charges, and Functions. Radon-Nikodym Theorem. Change of Variables in the Lebesgue Integral
If a function f is summable over a measure μ, then, according to Theorem 3.5.4, the charge ρ(E) = ∫ E fdμ is absolutely continuous with respect to the measure μ. Here, we give a mor...
