Lecture notes to the course Functional Analysis 1
Winter semester 2023/2024
Lecture notes to the preceeding course (only in Czech)
Introduction to Functional Analysis (2022/2023)
| Appendix: Basic notions and results in topology - | Czech, English |
V. Locally convex spaces
| V.1 Locally convex topologies and their generating - | Czech, English |
A proof of Theorem V.4(2)
| V.3 Spaces of finite and infinite dimension | - Czech, English |
A proof of Proposition V.17
A proof of the implication (iii)⇒(i) of Theorem V.20
A proof of Proposition V.21(2)
A proof of Example V.24(2)
A proof of Theorem V.29
A proof of Theorem V.30
VI. Weak topologies
A proof of the nontrivial implication from Theorem VI.8
A proof Theorem VI.15
VII. Elements of the theory of distributions
| VII.1 Space of test functions and weak derivatives - | Czech, English |
Proofs of Lemmata VII.1 and VII.2
A proof of Proposition VII.3
A proof of Theorem VII.4
| VII.2 Distributions - basic properties and operations - | Czech, English |
A proof of Proposition VII.8(d)
A proof of Proposition VII.9
A proof of Proposition VII.12
A proof of Lemma VII.13
A proof of Proposition VII.14 (including Lemma R)
A proof of Theorem VII.15
On convolution of two distributions, a proof of Proposition VII.16
Remarks and proofs to Theorem VII.19
Remarks and proofs from Proposition VII.20
| VII.6 Convolutions and the Fourier transform
of tempered distributions - | Czech, English |
A proof of Lemma VII.24
A proof of Theorem VII.25
Proofs of Lemma VII.26 and Proposition VII.27 (including Lemma RT)
A proof of Theorem VII.28
VIII. Elements of vector integration
| VIII.1 Measurability of vector-valued functions - | Czech, English |
A proof of Propositition VIII.1
A proof of Lemma VIII.2
A proof of the implication (iii)⇒(i) in Theorem VIII.3
| VIII.2 Integrability of vector-valued functions - | Czech, English |
Proofs of Propositition VIII.7 and Theorem VIII.8
A proof Theorem VIII.14(a-c)
A proof Theorem VIII.15
Three solved problems (the first one was presented during the classes
IX. Compact convex sets - Czech, English
Proofs of Lemma IX.2 and Theorem IX.3
A proof of Proposition IX.4
Some examples on extreme points (including Example IX.5)
A proof of Proposition IX.6
Proofs of Proposition IX.7 and Theorem IX.8
Problems to Chapter IX
