Faculty of Mathematics and Physics

Functional Analysis 1

Information in Student Information System


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


Overview of exam results

Lecture notes to the course Functional Analysis 1

Winter semester 2021/2022


Introductory information -

Czech, English

Appendix: Basic notions and results in topology -

Czech, English


I. Topological vector spaces

I.1 Linear topologies and their generating -

Czech, English

        A proof of Theorem I.4(2)

I.2 Continuous and bounded linear mappings
I.3 Spaces of finite and infinite dimension

} - Czech, English

        A proof of Proposition I.9

        A proof of the implication (iii)⇒(i) of Theorem I.11

I.4 Metrizability of TVS
I.5 Minkowski functionals, seminorms
  and generating of locally convex topologies

} - Czech, English

        A proof of Proposition I.13

        A proof of Lemma I.16

        A proof of Proposition I.23(2)

I.6 F-spaces and Fréchet spaces -

Czech, English

        A proof of three cases from Example I.26(3)

        A proof of Theorem I.31 (including the version for TVS)

        A proof of Theorem I.32

I.7 Separation theorems in locally convex spaces -

Czech, English


Problems to Chapter I -

Czech, English

        Analysis of the space ⋂p∈(0,∞)Lp(R).


II. Weak topologies

II.1 General weak topologies and duality -

Czech, English

II.2 Weak topologies on LCS
II.3 Polars and their applications

} - Czech, English

        A proof of the nontrivial implication from Theorem II.8

        A proof Theorem II.15


Problems to Chapter II -

Czech, English


III. Elements of vector integration

III.1 Measurability of vector-valued functions -

Czech, English

        A proof of Propositition III.1

        A proof of Lemma III.2

        A proof of the implication (iii)⇒(i) in Theorem III.3

III.2 Integrability of vector-valued functions -

Czech, English

        A proof of Propositition III.7 and Theorem III.8

        A supplement on convergence of series in normed spaces (in Czech)

                A proof of remark (1) (in Czech)

                A proof of Proposition 27 (in Czech)

                A proof of Proposition 29(a) (in Czech)

III.3 Lebesgue-Bochner spaces -

Czech, English

        A proof Theorem III.14(a-c)

        A proof Theorem III.15


Problems to Chapter III -

Czech,English

        Three solved problems from the classes


IV. Banach algebras and Gelfand transform

IV.1 Basic notions and properties -

Czech, English

        A proof of Proposition IV.2

        A proof of Lemma IV.6

        A proof of Theorem IV.7

IV.2 Spectrum and its properties -

Czech, English

        Comparison of invertibility and spectrum in A and in A+

        A proof of Proposition IV.8

        Proofs of Theorem IV.9 - Lemma IV.11

        A proof of Theorem IV.12

        A proof of Proposition IV.14 and Corollary IV.15

IV.3 Holomorphic functional calculus -

Czech, English

        A proof of Proposition IV.16

        A proof of Theorem IV.17 and of the related remarks

IV.4 Ideals, complex homomorphisms
    and Gelfand transform

- Czech, English

        A proof of Proposition IV.21

        A proof of Proposition IV.22

        A proof of Proposition IV.23

        A proof of Theorem IV.24 (including the preceding definitions)

IV.5 C*-algebras - basic properties -

Czech, English

        A proof of Proposition IV.29

        A proof of Example IV.31

        A proof of Proposition IV.32

        A proof of Theorem IV.24 and of Corollary IV.34

        A proof of Corollary IV.35

IV.6 Continuous functional calculus in C*-algebras -

Czech, English

        A proof of Proposition VIII.36

        A proof of Theorem VIII.37

        A proof of Theorem VIII.38

        A proof of Theorem VIII.39


Problems to Chapter IV -

Czech,English