Functional analysis problems

[1379/32%]

P. A. Borodin, A. M. Savchuk, I. A. Sheypak

Chapter 1Metric spaces[74/32%]

Section 1.1Main concepts and properties[18/39%]

Section 1.2Sequences in metric spaces. Completeness[21/38%]

Section 1.3Everywhere dense sets. Baire's theorem[17/18%]

Section 1.4Metric space mappings[8/25%]

Section 1.5Fixed Point Theorem[10/40%]

Chapter 2Normed spaces[98/29%]

Section 2.1Main concepts and properties. Examples of normed spaces[18/17%]

Section 2.2Sets and sequences in normed spaces. Subspaces[26/50%]

Section 2.3Banach spaces[32/22%]

Section 2.4Constructions of Banach spaces. Direct sums of subspaces[12/25%]

Section 2.5Separability of normed spaces[10/20%]

Chapter 3Hilbert spaces[71/31%]

Section 3.1Main concepts and properties. Examples of Euclidean and Hilbert spaces[20/5%]

Section 3.2Sets in Hilbert spaces[24/46%]

Section 3.3Orthogonal and orthonormal systems and bases in Hilbert spaces[27/37%]

Chapter 4Compact sets[63/32%]

Section 4.1Properties of compact sets[38/16%]

Section 4.2Compact sets in specific normed spaces[25/56%]

Chapter 5Linear continuous functionals[81/32%]

Section 5.1Main properties. Calculation of norms[23/26%]

Section 5.2Hahn—Banach Theorem[17/18%]

Section 5.3Dual spaces[26/54%]

Section 5.4The second adjoint space. Reflexivity[15/20%]

Chapter 6Linear operators[100/46%]

Section 6.1Definitions and basic examples of operators[41/59%]

Section 6.2Different properties of operators[20/35%]

Section 6.3Operators in Hilbert spaces[15/40%]

Section 6.4Operator space[15/40%]

Section 6.5Differentiation in Banach spaces[9/33%]

Chapter 7Banach-Steinhaus Theorem. Weak convergence of vectors, functionals, and operators[78/24%]

Section 7.1Banach-Steinhaus Theorem[6/17%]

Section 7.2Weak convergence: basic properties. Criteria for weak convergence[30/23%]

Section 7.3*-weak convergence in the dual space[22/23%]

Section 7.4Different types of convergence in the space of operators[20/30%]

Chapter 8Adjoint operators[81/41%]

Section 8.1Adjoint operators in a Banach space[14/50%]

Section 8.2Adjoint operators in a Hilbert space. Unitary and normal operators[67/39%]

Chapter 9Inverse operator[59/41%]

Section 9.1Banach's Theorem on the Inverse Operator. Examples[35/51%]

Section 9.2Properties of invertible operators[24/25%]

Chapter 10Bases[64/22%]

Section 10.1Complete and minimal systems of vectors[20/35%]

Section 10.2Shauder bases[32/22%]

Section 10.3Bases in Hilbert spaces[12/0%]

Chapter 11Compact operators and Fredholm theory[100/30%]

Section 11.1General properties of compact operators[29/21%]

Section 11.2Compact operators in specific spaces[19/26%]

Section 11.3Compact operators in Hilbert spaces[16/25%]

Section 11.4Fredholm theory[25/20%]

Section 11.5Integral equations[11/91%]

Chapter 12Fundamentals of spectral theory of bounded operators in Banach spaces[111/32%]

Section 12.1Spectrum[58/29%]

Section 12.2The spectrum of a compact operator[19/47%]

Section 12.3Hilbert–Schmidt theorem[32/22%]

Section 12.4Main types of operators illustrated by examples[2/100%]

Chapter 13Functional calculus and the spectral theorem[89/37%]

Section 13.1Functional calculus of a bounded operator[16/56%]

Section 13.2Functional calculus constructed by a self-adjoint operator[21/19%]

Section 13.3Spectral theorem in terms of the Lebesgue–Stieltjes integral[25/32%]

Section 13.4Spectral theorem in terms of the multiplication operator[27/44%]

Chapter 14Topological, linear topological and polynormed spaces[129/21%]

Section 14.1Topological spaces[31/32%]

Section 14.2Linear topological spaces[21/33%]

Section 14.3Locally convex spaces as polynormed spaces[43/12%]

Section 14.4The weak topology in a normed space[14/21%]

Section 14.5*-weak topology in the dual space[20/10%]

Chapter 15Test (basic) function spaces[34/26%]

Chapter 16Generalized functions[52/35%]

Section 16.1Main concepts[26/35%]

Section 16.2Operations on generalized functions[26/35%]

Chapter 17Fourier transform[40/28%]

Section 17.1Fourier transform of ordinary functions[33/24%]

Section 17.2Fourier transform of generalized functions[7/43%]

Chapter 18Convolution[37/30%]

Section 18.1Convolution of functions in

L^1(\mathbb{R})

[15/27%]

Section 18.2Convolution operator in

L^2(\mathbb{R})

[5/60%]

Section 18.3Convolution of generalized functions[17/24%]

Chapter 19Generalized functions of several variables[18/67%]

Section 19.1Additional operations on generalized functions[8/75%]

Section 19.2Fundamental solutions[10/60%]