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

Granas, A.

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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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A. Granas

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Rozprawy Matematyczne tom/nr w serii: 30 wydano: 1962

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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

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Instytut Matematyczny Polskiej Akademii Nauk

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Warszawa

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Rozprawy Matematyczne tom/nr w serii: 30

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93

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Rozprawy Matematyczne, Tom XXX

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1962

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autor

A. Granas

Bibliografia

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[3] Altman, M., Continuous transformations of open sets in locally convex spaces, Bull. Acad. Polon. Sci. 6 (1958), pp. 297-301.
[4] Birkhoff, G. D., (with O. D. Kellogg) Invariant points in function spaces, Trans. Amor. Math. Soc. 23, (1922), pp. 96-115.
[5] Borsuk, K., Sur un espace des transformations continues et ses applications topologiques, Monatsh. für Math. u. Phys., 38, (1931), pp. 381-386.
[6] Borsuk, K., Über Schnitte der n-dimensionalen Euklidischen Räume, Math. Ann. 106 (1932), pp. 239-248.
[7] Borsuk, K., Drei Satze über die n-dimensionale Euklidische Sphäre, Fund. Math. 21 (1933), pp. 177- 190.
[8] Borsuk, K., Über stetige Abbildungen der Euklidischen Räume, Fund. Math. 21 (1933), pp. 236-243.
[9] Borsuk, K., Sur les prolongements des transformations continues, Fund. Math. 28 (1937), pp. 99-110.
[10] Browder, F. E., On a generalization of the Schauder Fixed Point Theorem, Duke Math. Journ. 26 (1959), pp. 291-304.
[11] Browder, F. E., Non-linear functional equations in locally convex spaces, Duke Math. Journ. 24 (1957), pp. 579-590.
[12] Caccioppoli, R., Sugli elementi uniti delle transformazioni funzionali, Rend. Sem. Mat. Univ. Padova (1932), pp. 3, 1-15.
[13] Dugondji, J., An extension of Tietze's theorem, Pacific Journ. Math. 1 (1951), pp. 353-367.
[14] Fan, Ky, A generalization of Tucker's combinatorial lemma with topological applications, Ann. of Math. 56 (1952), pp. 431-437.
[15] Fan, Ky, Combinatorial properties of certain simplicial and cubical vertex maps, Arch. Math. 11 (1960), pp. 368-377.
[16]Gęba, K., (with A. Granas and A. Jankowski,) Some theorems on the sweeping in Banach spaces, Bull. Acad. Polon. Sci. 9 (1959), pp. 539-544.
[17] Granas, A., On a certain class of non-linear mappings in Banach spaces, (Russian with German summary) Bull. Acad. Polon. Sci., Cl. III, 9 (1957), pp. 867-871.
[18] Granas, A., On a certain geometrical theorem in Banach spaces, (Russian with German summary) Bull. Acad. Polon. Sci., Cl. III, 9 (1957), pp. 873-877.
[19] Granas, A., Über einen Satz von K. Borsuk, Bull. Acad. Polon. Sci., Cl. III, 10 (1957), pp. 959-961.
[20] Granas, A., On continuous mappings of open sets in Banach spaces, (Russian with English summary) Bull. Acad. Polon. Sci. 1 (1958), pp. 25-30.
[21] Granas, A., Homotopy Extension Theorem in Banach spaces and some of its applications to the theory of non-linear equations, (Russian with English summary) Bull. Acad. Polon. Sci. 7 (1959), pp. 387-394.
[22] Granas, A., See Gęba, К. [1].
[23] Granas, A., Theorem on Antipodes and Theorems on Fixed Points for a certain class of multivalued mappings in Banach spaces, Bull. Acad. Polon. Sci. 8 (1959), pp. 271-275.
[24] Granas, A., Sur la notion de degré topologique pour une certaine classe de transformations multivalentes dans les espaces de Banach, Bull. Acad. Polon. Sci. 7 (1959), p. 191.
[25] Granas, A., On the disconnection of Banach spaces, Fund. Math. 48 (1960), pp. 189-200.
[26] Granas, A., Introduction to topology of functional spaces, Lecture Notes, University of Chicago (1961).
[27] Granas, A., An extension to functional spaces of Borsuk's-Ulam Theorem on Antipodes, Bull. Acad. Polon. Sci. 10 (1962), pp. 81-86.
[28] Granas, A., A note on compact deformation in functional spaces. Bull. Acad. Polon. Sci. 10 (1962), pp. 87-90.
[29] Granas, A., Sur la multiplication cohomotopique dans les espaces de Banach, C. R. Acad. Sci. Paris 254 (1962), pp. 56-57.
[30] Granas, A., A note on Schauder's Theorem on invariance of domain, (Russian with English summary), Bull. Acad. Polon. Sci. 10 (1962), pp. 233-238.
[31] Jankowski, A., See Gęba, К. [1].
[32] Kakutani, S., Topological properties of the unit sphere of a Hilbert Space, Proc. Imp. Acad. Tokyo 19 (1943), pp. 269-271.
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[34] Klee, V., Convex bodies and periodic homeomorphisms in Hilbert space, Trans. Amer. Math. Soc. 74 (1953), pp. 10-43.
[35] Klee, V., Leray-Schauder theory without local convexity, Math. Ann. 141 (1960), pp. 286-296.
[36] Kolmogoroff, A. N., (with S. V. Fomin) Elements of the theory of functions and functional analysis, (in Russian) Moscow 1954. English translation: Graylock Press, Rochester, N. Y. 1957.
[37] Krasnoselskiĭ, M. A., Topological methods in the theory of non-linear equations, (in Russian) Gostechizdat, Moscow (1956).
[38] Krasnoselskiĭ, M. A., On the theory of completely continuous fields, (in Russian) Ukrain. Mat. Ž. 3 (1951), pp. 174-183.
[39] Krasnoselskiĭ, M. A., (with S. Krein) On a proof of a theorem on the category of a projective space, (in Russian) Ukrain. Math. Ż. 1 (1949), pp. 99-102.
[40] Krasnoselskiĭ, M. A., (with L. Ladyzenskiĭ) Structure of the spectrum of positive nonhomogeneous operators, (in Russian) Trudy Moskov. Mat. Obsc. 3 (1954), pp. 321-346.
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[42] Kuratowski, K., Introduction to set theory and topology, Państwowe Wydawnictwo Naukowe, Warszawa 1961.
[43] Kyner, W. T., A generalization of the Borsuk and Borsuk-Ulam Theorem, Proc. Amer. Math. Soc. 7 (1956), pp. 1117-1119.
[44] Lefschetz, S., Introduction to topology, Princeton University Press, Princeton 1949. Leray, J.
[45] Leray, J., (with J. Schauder) Topologie et équations fonctionelles, Ann. Sci. Ecole Norm. Sup. 51 (1934), pp. 45-78.
[46] Leray, J., Topologie des espaces abstraits de M. Banach, C. R. Acad. Sci. Paris 200 (1935), pp. 1083-1093.
[47] Leray, J., Les problèmes non linéaires, L’Enseignement mathématique 35 (1930), pp. 139-151.
[48] Leray, J., Sur la position d'un ensemble fermé de points d’un espace topologique, J. Math. Pures et Appl., 9e série 24 (1945), pp. 169-199.
[49] Leray, J., Sur les équations et les transformations, Jour, de Math. Pures et Appl., 9e série, 24 (1945), pp. 201-248.
[50] Leray, J., La théorie des points fixes et ses applications en analyse, Proc. Int. Congress Math., Cambridge, Mass., 2 (1950), 202-208.
[51] Leray, J., Théorie des points fixes: indice total et nombre de Lefschetz, Bull. Soc. Math. France 87 (1959), pp. 221-233.
[52] Lusternik, L., (with L. Schnirelman) Topological methods in variational calculus, (in Russian) Moscow 1930.
[53] Lusternik, L., (with Sobolev) Elements of Functional Analysis, (in Russian) Gostiechizdat, Moscow 1951. English translation: Gordon and Breach Scientific Publishers, New York 1961.
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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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