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

A. Granas

GDML_Books, (1962), p.

Résumé

§1-3 Lusternik [1] and Schnirelman, Borsuk [3]; see also Tucker [1], Krasnoselskiï [3] and Krein, Fan Ky [1, 2], Lefshetz [1].

TABLE OF CONTENTSINTRODUCTION................................................................................................................................................................................................... 3PRELIMINARIES1. Metric spaces.................................................................................................................................................................................................... 52. Normed and Banach spaces......................................................................................................................................................................... 12CHAPTER I. Extension problems1. Extension of mappings. Tietze’s Extension Theorem.............................................................................................................................. 172. Homotopy, retraction and fixed point property............................................................................................................................................. 193. Essential and inessential mappings. Borsuk’s Antipodensatz and Brouwer’s Fixed Point Theorem........................................... 20CHAPTER II. Compact and finite dimensional mappings1. Approximation Theorem.................................................................................................................................................................................. 232. Examples of compact mappings.................................................................................................................................................................. 263. Extension of compact mappings................................................................................................................................................................... 28CHAPTER III. Compact vector fields and Homotopy Extension Theorem1. The space $(\mathfrak {C}(Y^X)$. Singularity free compact fields........................................................................................................... 322. Homotopy of compact vector fields............................................................................................................................................................... 343. Extension of compact fields and the Homotopy Extension Theorem.................................................................................................... 37CHAPTER IV. Essential and inessential compact fields. Theorems on Antipodes1. Essential and inessential compact fields. Schauder Fixed Point Theorem......................................................................................... 392. The First Theorem on Antipodes in Banach spaces................................................................................................................................ 413. The Second Theorem on Antipodes............................................................................................................................................................. 434. Alternative of Fredholm.................................................................................................................................................................................... 46CHAPTER V. Continuous continuation method and fixed-point theorems1. Continuous continuation method.................................................................................................................................................................. 482. Theorems on fixed points............................................................................................................................................................................... 50CHAPTER VI. Compact deformations. Theorem on the Sweeping. Birkhoff-Kellogg Theorem1. Separation between two points. Theorems on compact deformations................................................................................................ 542. Birkhoff-Kellogg Theorem.............................................................................................................................................................................. 563. Invariant directions for positive operators.................................................................................................................................................... 58

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