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The theory of compact vector fields and some of its applications to topology of functional spaces (I)

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Rozprawy Matematyczne tom/nr w serii: 30 wydano: 1962
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EN
§1-3 Lusternik [1] and Schnirelman, Borsuk [3]; see also Tucker [1], Krasnoselskiï [3] and Krein, Fan Ky [1, 2], Lefshetz [1].
EN

TABLE OF CONTENTS
INTRODUCTION................................................................................................................................................................................................... 3
PRELIMINARIES
1. Metric spaces.................................................................................................................................................................................................... 5
2. Normed and Banach spaces......................................................................................................................................................................... 12
CHAPTER I. Extension problems
1. Extension of mappings. Tietze's Extension Theorem.............................................................................................................................. 17
2. Homotopy, retraction and fixed point property............................................................................................................................................. 19
3. Essential and inessential mappings. Borsuk's Antipodensatz and Brouwer's Fixed Point Theorem........................................... 20
CHAPTER II. Compact and finite dimensional mappings
1. Approximation Theorem.................................................................................................................................................................................. 23
2. Examples of compact mappings.................................................................................................................................................................. 26
3. Extension of compact mappings................................................................................................................................................................... 28
CHAPTER III. Compact vector fields and Homotopy Extension Theorem
1. The space $(\mathfrak{C}(Y^X)$. Singularity free compact fields........................................................................................................... 32
2. Homotopy of compact vector fields............................................................................................................................................................... 34
3. Extension of compact fields and the Homotopy Extension Theorem.................................................................................................... 37
CHAPTER IV. Essential and inessential compact fields. Theorems on Antipodes
1. Essential and inessential compact fields. Schauder Fixed Point Theorem......................................................................................... 39
2. The First Theorem on Antipodes in Banach spaces................................................................................................................................ 41
3. The Second Theorem on Antipodes............................................................................................................................................................. 43
4. Alternative of Fredholm.................................................................................................................................................................................... 46
CHAPTER V. Continuous continuation method and fixed-point theorems
1. Continuous continuation method.................................................................................................................................................................. 48
2. Theorems on fixed points............................................................................................................................................................................... 50
CHAPTER VI. Compact deformations. Theorem on the Sweeping. Birkhoff-Kellogg Theorem
1. Separation between two points. Theorems on compact deformations................................................................................................ 54
2. Birkhoff-Kellogg Theorem.............................................................................................................................................................................. 56
3. Invariant directions for positive operators.................................................................................................................................................... 58
Słowa kluczowe
Tematy
Miejsce publikacji
Warszawa
Copyright
Seria
Rozprawy Matematyczne tom/nr w serii: 30
Liczba stron
93
Liczba rozdzia³ów
Opis fizyczny
Rozprawy Matematyczne, Tom XXX
Daty
wydano
1962
Twórcy
autor
Bibliografia
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