Ondřej Kalenda, KMA MFF UK

Lecture notes to the course Functional Analysis 1

Winter semester 2016/2017

Lecture notes to the preceeding course (only in Czech)

Introduction to Functional Analysis (2015/2016)

Introductory information -

Czech, English

Appendix: Basic notions and results in topology -

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V. Topological vector spaces

V.1 Linear topologies and their generating -

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A proof of Theorem V.4(2)

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

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

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

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A proof of Proposition V.13

A proof of Lemma V.16

A proof of the implication (iv)⇒(ii) of Proposition V.22

V.6 F-spaces and Fréchet spaces -

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A proof of three cases from Example V.25(3)

A proof of Theorem V.31

V.7 Separation theorems in locally convex spaces -

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Problems to Chapter V -

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VI. Weak topologies

VI.1 General weak topologies and duality -

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

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A proof of the nontrivial implication from Theorem VI.8

A proof Theorem VI.15

Problems to Chapter VI -

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VII. Elements of vector integration

VII.1 Measurability of vector-valued functions -

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A proof of Propositition VII.1

A proof of Lemma VII.2

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

VII.2 Integrability of vector-valued functions -

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

VII.3 Lebesgue-Bochner spaces -

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A proof Theorem VII.14(a-c)

A proof Theorem VII.15

Problems to Chapter VII -

here

VIII. Banach algebras and Gelfand transform

VIII.1 Basic notions and properties -

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A proof of Proposition VIII.2

A proof of Lemma VIII.6

A proof of Theorem VIII.7

VIII.2 Spectrum and its properties -

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Comparison of invertibility and spectrum in A and in A⁺

A proof of Proposition VIII.8

Proofs of Theorem VIII.9 - Theorem VIII.11

A proof of Theorem VIII.12

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

VIII.3 Holomorphic functional calculus -

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A proof of Proposition VIII.16

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

VIII.4 Ideals, complex homomorphisms
and Gelfand transform

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A proof of Proposition VIII.21

A proof of Proposition VIII.22

A proof of Proposition VIII.23

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

VIII.5 C*-algebras - basic properties -

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A proof of Proposition VIII.29

A proof of Example VIII.31

A proof of Proposition VIII.32

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

A proof of Corollary VIII.35

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

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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 VIII (currently to Sections VIII.1-VIII.3) -

here

IX. Operators on a Hilbert space

IX.1 Various types of operators and their properties

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A proof of Proposition IX.1

A proof of Proposition IX.2

A proof of Lemma IX.3 and Proposition IX.4

A proof of Proposition IX.5

A proof of Proposition IX.6

A proof of Proposition IX.7

A proof of Theorem IX.8

A proof of Theorem IX.9, Proposition IX.10 and Theorem IX.11

IX.1 Measurable calculus and spectral decomposition

- English

A proof of Proposition IX.12

From the definition of the spectral measure to Lemma IX.14

A proof of Theorem IX.15

A proof of Lemma IX.16

From Proposition IX.17 to Corollary IX.20

Ondrej Kalenda

Department of Mathematical Analysis

Faculty of Mathematics and Physics

Charles University in Prague

Functional Analysis 1

Lecture notes to the course Functional Analysis 1

Winter semester 2016/2017