Faculty of Mathematics and Physics

Functional Analysis 1

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Organization of the course


Content of the course, expected knowledge
and connections to other courses


Recommended literature


Lecture notes


Credit requirements
(and statistical data on homeworks)


Content of the lectures and classes


Conditions for exams (examination has been finished)


Overview of exam results

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)


Introductory information -

Czech, English

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.2 Continuous and bounded linear mappings

- Czech, English

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

V.4 Metrizability of locally convex spaces

Czech, English

        A proof of Proposition V.21(2)

V.5 Fréchet spaces -

Czech, English

        A proof of Example V.24(2)

        A proof of Theorem V.29

        A proof of Theorem V.30

V.6 Extension and separation theorems -

Czech, English


Problems to Chapter V -

Czech, English


VI. Weak topologies

VI.1 General weak topologies and duality -

- Czech, English

VI.2 Weak topologies on LCS

- Czech, English

        A proof of the nontrivial implication from Theorem VI.8

VI.3 Polars and their applications

- Czech, English

        A proof Theorem VI.15


Problems to Chapter VI -

Czech, English


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

VII.3 A bit more on distributions -

Czech, English

        A proof of Proposition VII.12

VII.4 Convolutions of distributions -

Czech, English

        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

VII.5 Tempered distributions -

Czech, English

        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


Problems to Chapter VII -

Czech,English


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

VIII.3 Lebesgue-Bochner spaces -

Czech, English

        A proof Theorem VIII.14(a-c)

        A proof Theorem VIII.15


Problems to Chapter VIII -

Czech,English

        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