Lecture
INTRODUCTION
CHAPTER 1. TOPOLOGICAL SPACES
1. The concept of a set. Operations on sets. Mappings. The characteristic function of a set
2. Topology and topological space. Base of a topology
3. The structure of open sets and neighbourhoods
4. The concept of a metric space and the topology defined by a metric. Examples of metric spaces
5. The closure operation of a set in a topological space
6. Interior points of a set, interior. Boundary of a set
7. Separable topological spaces
8. Induced topologies and quotient topology
9. Continuous mapping. Homeomorphism
10. Compact spaces
CHAPTER 2. PROPERTIES OF METRIC SPACES
1. Convergent sequences in metric spaces and complete metric spaces
2. The theorem on the completion of a metric space
3. The criterion for completeness of a space
4. Compact sets in a metric space. The Hausdorff theorem
5. Criteria for compactness in the spaces C[0, 1], lp. The Arzelà theorem
6. The Weierstrass theorem on uniform approximation and separability of C[0, 1]
7. Mapping of compact sets. The Weierstrass theorem on the boundedness and attainment of exact bounds by a continuous function
8. The contraction mapping principle and its application
9. Nowhere dense sets. The concept of category of sets of a metric space. The Baire theorem
CHAPTER 3. MEASURE AND MEASURABLE SETS
1. Systems of sets
2. Systems of sets in Euclidean space
3. Set function
4. Measure and its simplest properties. Measure in Euclidean space
5. Outer measure
6. Measurable sets
7. Lebesgue measure on Rn
CHAPTER 4. MEASURABLE FUNCTIONS
1. Measurable functions and their properties
2. Convergence almost everywhere
3. Convergence in measure and its properties
4. Comparison of convergence almost everywhere and in measure
5. Almost uniform convergence. The theorems of Egorov and Luzin
CHAPTER 5. THE LEBESGUE INTEGRAL
1. The Lebesgue integral for simple and bounded functions on a space with finite measure
2. The basic properties of the integral of a bounded function
3. Definition of the Lebesgue integral in the general case
4. Passing to the limit under the integral sign
5. Comparison of the Riemann and Lebesgue integrals
6. Charges. The Radon—Nikodym theorem
7. σ-additivity of the direct product of measures. The Fubini theorem
CHAPTER 6. NORMED AND HILBERT SPACES
1. Linear spaces. Normed spaces. The metric induced by a norm. Series in normed spaces. Absolute convergence of a series and completeness of a normed space. Quotient spaces
2. Finite-dimensional spaces. Finite dimensionality and compactness. The Riesz theorem on local compactness.
3. Inner product. Hilbert space. Axioms and properties. Orthonormal systems. Schmidt orthogonalization. The parallelogram identity.
4. Orthogonality and orthogonal complement
5. Fourier series in Hilbert space. Fourier coefficients. Bessel's inequality and Parseval's equality. Complete and closed orthonormal systems
CHAPTER 7. LINEAR OPERATORS IN NORMED SPACES
1. Linear operators in linear normed spaces. The equivalence of continuity and boundedness of a linear operator. The concept of the norm of a bounded operator. Various formulas for computing norms. Examples of bounded linear operators.
2. The space of linear continuous operators and its completeness with respect to uniform convergence of operators
3. The principle of uniform boundedness and the Banach–Steinhaus theorem. Completeness of the space of operators with respect to pointwise convergence
4. The kernel of an operator. The criterion for boundedness of the inverse operator. Theorems on the inverse operator
5. Examples of inverse operators. Invertibility of operators of the form (I - A) and (A - C).
6. The graph of an operator and closed operators. The criterion for closedness. The Banach closed graph theorem. The open mapping theorem
CHAPTER 8. LINEAR FUNCTIONALS IN NORMED SPACES
1. Linear continuous functionals. Extension by continuity. The Hahn–Banach theorem. Corollaries of the Hahn–Banach theorem
2. Dual spaces
3. The Riesz theorem on the general form of a linear functional for the space of continuous functions
4. Lebesgue spaces and their duals
5. Isomorphism and isometry of separable Hilbert spaces. The general form of a linear functional in a Hilbert space. The Riesz–Fischer theorem.
6. The adjoint operator. Conditions for the existence of an adjoint operator. Closedness of the adjoint operator. The adjoint of a bounded operator and its norm.
7. The self-adjoint operator. The norm of a self-adjoint operator
CHAPTER 9. SPECTRAL THEORY OF OPERATORS
1. Completely continuous operators and their properties. Fredholm and Hilbert–Schmidt operators
2. The Schauder theorem on the complete continuity of the adjoint operator. Equations of the first and second kind with completely continuous operators. The theorem on the closedness of the range of an operator
3. The Fredholm alternative. The Schauder fixed point theorem.
4. The resolvent and spectrum of an operator. Linear independence of eigenvectors. The spectrum of a completely continuous operator (finite dimensionality of the eigenspace, a finite number of eigenvalues outside a circle)
5. Existence of an eigenvalue for a completely continuous operator in a Hilbert space. The largest and smallest eigenvalues. Spectral decomposition of self-adjoint operators. The Hilbert–Schmidt theorem on expansion in eigenvectors
SUBJECT INDEX
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