Faculty of Mathematics and Physics

Functional Analysis 2

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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