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

100%

Granas A.

Instytut Matematyczny Polskiej Akademii Nauk

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

2

On local disconnection of Euclidean spaces

31%

Granas A.

Fundamenta Mathematicae

|

1955

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tom 41

|

nr 1

42-48

3

On the disconnection of Banach spaces

25%

Granas A.

Fundamenta Mathematicae

|

1959-1960

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tom 48

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nr 2

189-200

Ograniczanie wyników

2 Fundamenta Mathematicae

3 Granas A.

1 1962

1 1960

1 1955

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

On local disconnection of Euclidean spaces

On the disconnection of Banach spaces