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

A. Granas

  • Publisher: Instytut Matematyczny Polskiej Akademi Nauk(Warszawa), 1962

Abstract

top
§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 ( ( 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

How to cite

top

A. Granas. The theory of compact vector fields and some of its applications to topology of functional spaces (I). Warszawa: Instytut Matematyczny Polskiej Akademi Nauk, 1962. <http://eudml.org/doc/268396>.

@book{A1962,
abstract = {§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},
author = {A. Granas},
keywords = {functional analysis},
language = {eng},
location = {Warszawa},
publisher = {Instytut Matematyczny Polskiej Akademi Nauk},
title = {The theory of compact vector fields and some of its applications to topology of functional spaces (I)},
url = {http://eudml.org/doc/268396},
year = {1962},
}

TY - BOOK
AU - A. Granas
TI - The theory of compact vector fields and some of its applications to topology of functional spaces (I)
PY - 1962
CY - Warszawa
PB - Instytut Matematyczny Polskiej Akademi Nauk
AB - §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
LA - eng
KW - functional analysis
UR - http://eudml.org/doc/268396
ER -

Citations in EuDML Documents

top
  1. Marián J. Fabián, Concerning interior mapping theorem
  2. Antoni L. Dawidowicz, Krystyna Twardowska, On random boundary value problems for ordinary differential equations
  3. Josef Kolomý, Some existence theorems for nonlinear problems
  4. M. Furi, M. Martelli, A degree for a class of acyclic-valued vector fields in Banach spaces
  5. Josef Kolomý, The solvability of non-linear integral equations
  6. Andrzej Granas, The Leray-Schauder index and the fixed point theory for arbitrary ANRs
  7. Siegfried Hahn, Eigenwertaussagen für kompakte und kondensierende mengenwertige Abbildungen in topologischen Vektorräumen
  8. Siegfried Hahn, Gebietsinvarianzsatz und Eigenwertaussagen für konzentrierende Abbildungen
  9. Siegfried Hahn, Ein Homotopieerweiterungssatz für kompakte Vektorfelder in topologischen Vektorräumen (Vorläufige Mitteilung)
  10. Andrzej Granas, Topics in infinite dimensional topology

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.