Content of the lectures and classes
Lecture 1 - October 3, 2023
Introductory information - a brief content of the course, assumed knowledge, recommended literature etc.
Brief review of the necessary knowledge from general topology. Beginning of Section V.1 - definitions of topological vector spaces
and locally convex spaces.
Class 1 - October 3, 2023
Information on credit, Examples V.1(1)-(3) (only a part of example (3) - the metric, continuity of addition, characterization using locally
uniform convergence)
Lecture 2 - October 6, 2023
Section V.1 - completing Example V.1(3) (continuity of multiplication and local convexity), a part of example (5) (the metric, it is a HTVS),
continuation till Proposition V.3.
Lecture 3 - October 10, 2023
Continuation of Section V.1 - from Theorem V.4 to Lemma V.7. Example V.6(4) was omitted. A proof of Theorem V.4(2),
where some things were not clear during the lecture, is available at the lecture notes.
Class 2 - October 10, 2023
Student presentations - Problems 38, 39 and 40 to Chapter V. Problem 1 to Chapter V.
Lecture 4 - October 13, 2023
Completing of Section V.1 - a proof of continuity of multiplication in Theorem V.4(2), then from Proposition V.8 to the end of the section.
Beginning of Section V.2 (continuous and bounded linear mappings) - Proposition V.12 and the first part of Proposition V.13.
Lecture 5 - October 17, 2023
Completing of Section V.2 - the second part of Proposition V.13 and then continuation to the end of the section. Beginning of Section V.3
(spaces of finite and infinite dimension) - Proposition V.17 and Corollary V.18.
Class 3 - 13.10.2021
Student presentations - problems 19, 20, 21, 22 to Chapter V. Continuous linear functionals on C(R,F).
Lecture 6 - October 20, 2023
Completing of Section V.3 - from the definition of a totally bounded set to the end of the section. Beginning of Section V.4
(metrizability of locally convex spaces) - Proposition V.21.
Lecture 7 - October 24, 2023
Completing Section V.4 - Theorems V.23 and V.24. Beginning of Section V.5 (Fréchet spaces) - till Theorem V.29 (only a part of the
proof has been done).
Class 4 - October 24, 2023
Student presentations - problems 17, 18, 23, 24 to Chapter V. Description of continuous linear functionals on FΓ
and characterization of bounded sets in FN and in C(R,F).
Lecture 8 - October 27, 2023
Completion of Section V.5 - Theorems V.29 and V.30. Beginning of Section V.6 (extension and separation theorems) - till Corollary V.34.
Lecture 9 - October 31, 2023
Completion of Section V.6 (Theorem 35 and Corollary 36);
beginning of Chapter VI (weak topologies), especially of Section VI.1 (general weak topologies and duality) -
till Example 2(1).
Class 5 - October 31, 2023
Precision and correction of Problem 23 to Chapter V. Problems 1(3,4), 26 and 27 to Chapter V.
Lecture 10 - November 3, 2023
Completion of Section VI.1 (From Example VI.2(2) to the end of the section);
Section VI.2 (weak topologies on LCS) - to Theorem VI.8.
Lecture 11 - November 7, 2023
Completing Section VI.2 - Proposition VI.9. Beginning of Section VI.3 (polars and their application) - to Theorem VI.14.
Class 6 - November 7, 2023
Student presentations - problem 41 to Chapter V, problems 10, 11, 13, 14 to Chapter VI.
Lecture 12 - November 10, 2023
Completion of Section VI.3 (from Theorem VI.15 to the end of the section);
beginning of Chapter VII (elements of the theory of distributions) and Section VII.1 (space of test functions and weak derivatives)
- introductory definitions and remarks.
Lecture 13 - November 14, 2023
Completion of Section VII.1 - from Lemma VII.1 to the end of the section. In Theorem VII.4 the second part of (b) and assertion (c) were just
briefly commented (a detailed proof is available at lecture notes). Beginning of Section VII.2 (distributions - basic properties and operations)
- to Lemma VII.5, assertion (b) has not been proved yet.
Class 7 - November 14, 2023
Student presentations - problems 16, 17, 18 to Chapter VI.
Lecture 14 - November 21, 2023
Continuation of Section VII.2 - from Lemma VII.5(b) to Proposition VII.8.
Class 8 - November 21, 2023
Student presentations - problems 21, 22, 27, 29, 30 to Chapter VI. Additional comments - the norm and weak topologies coincide just in spaces
of finite dimension, in ℓ1 the origin belongs to the weak closure of the sphere, but it cannot be reached by a sequence,
canonical vectors in ℓ1(Γ) form a closed discrete set in the weak topology.
Lecture 15 - November 24, 2023
Completion of Section VII.2 - Proposition VII.9 (the second part of the proof of (b) was only sketched, a detailed proof is available
at the lecture notes ).
Beginning of Section VII.3 (a bit more on distributions) - to Proposition VII.12(a,b).
Lecture 16 - November 28, 2023
Completion of Section VII.3 - Proposition VII.12(c-e), proof of (e) was not done, just the first step was shown, a detailed proof is available at the lecture notes. Beginning of Section VII.4 (convolution of distributions) - to Lemma VII.13.
Class 9 - November 28, 2023
Student presentation - Problem 19 to Chapter VII. Problems 7 and 15 to Chapter VII.
Lecture 17 - December 1, 2023
Continuation of Section VII.4 - Proposition VII.14 and Theorem VII.15(a)-(d). Some parts of proofs were shown in a brief version, detailed versions are available at the lecture notes.
Lecture 18 - December 5, 2023
Completion of Section VII.4 - from Theorem VII.15(e) to the end of the section. Possibilities of defining the convolution of two distributions
were only briefly explained, without detailed proofs. Proposition VII.16 was only briefly commented. Detailed explanations and proofs are available
at the lecture notes. Beginning of Section VII.5 (tempered distributions) - to Proposition VII.17(a).
Class 10 - December 5, 2023
Student presentations - problems 21 and 29 to Chapter VII. Problems 27, 16, 20, 40 (a part) to Chapter VII.
Lecture 19 - December 8, 2023
Continuing Section VII.5 - from Proposition VII.17(b) to Lemma VII.21 (a part). Some details of computations were skipped, they are available at the lecture notes.
Lecture 20 - December 12, 2023
Completion of Section VII.5 - the remaining part of Lemma VII.21 and continuation to the end of the section. Beginning of Section VII.6
(convolutions and Fourier transform of tempered distributions) - to Theorem VII.25(a).
Class 11 - December 12, 2023
Student presentation - Problems 38 and 39 to Chapter VII. Then Problems 41, 48, 49, 50, 26(1,4) to Chapter VII.
Lecture 21 - December 15, 2023
Completion of Section VII.6 - from Theorem VII.25(b) to the end of the section. Some proofs were skipped - those of Lemma VII.26(b), Proposition VII.27 and Theorem VII.28(a,c,e). They are analogous to the respective proof from Section VII.4 and may be found at the lecture notes. Beginning of Chapter VIII
(Elements of vector integration) and Section VIII.1 (measurability of vector-valued functions) - basic definitions and related remarks.
Lecture 22 - December 19, 2023
Continuation of Section VIII.1 - Proposition VIII.1, Lemma VIII.2 and Theorem VIII.3, definition of strong μ-measurability, a remark saying
that a function is strongly μ-measurable if and only if it equals almost everywhere to a strongly Σ-measurable function.
Class 12 - November 19, 2023
Student presentations - Problems 22, 44, 46(1,2) to Chapter VII. Problems 45 and 46(4) to Chapter VII.
Recalling of the final definition and remark from the last lecture, an explanation how Lemma VIII.4 and Theorem VIII.5 follow from Lemma VIII.2 and Theorem VIII.3 using that remark. Examples VIII.6(1,3).
Lecture 23 - January 5, 2024
Beginning of Section VIII.2 (integrability of vector-valued functions) - to the definition of Pettis integral.
Lecture 24 - January 9, 2024
Completion of Section VIII.2 - repeating definitions of Dunford and Pettis integrals, Proposition VIII.12 and then to the end of the section.
Beginning of Section VIII.3 (Lebesgue-Bochner spaces) - to Theorem VIII.15(a).
Class 13 - January 9, 2024
Completing Section VIII.3 - Theorem VIII.15(b), Examples VIII.16 and remarks on duality. An analysis of the function t↦ψχ(0,t) as function
(0,1)→Lp(0,1) (measurability, Bochner, weak and Pettis integrability, the value of the integral).
Lecture 25 - January 12, 2024
Chapter IX (Compact convex sets) - to Theorem IX.3, Proposition IX.4 was only briefly commented, Example IX.5 and Proposition IX.6,
brief info on Proposition IX.7 and Theorem IX.8.
