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Basics in Nonlinear Geometric Analysis

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Banach Space Theory

Part of the book series: CMS Books in Mathematics ((CMSBM))

Abstract

In this chapter, we begin by proving the Brouwer and the Schauder fixed-point theorems. Then we turn to results on homeomorphisms of convex sets and spaces. We prove Keller’s theorem on homeomorphism of infinite-dimensional compact convex sets in Banach spaces to \({\mathbb I}^{{\mathbb N}}\). We also prove the Kadec theorem on the homeomorphism of every separable reflexive space to a Hilbert space. Then we prove some results on uniform, in particular Lipschitz, homeomorphisms.

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References

  1. I. Aharoni, Uniform embeddings of Banach spaces, Israel J. Math. 27 (1977), 174–179.

    Article  MATH  MathSciNet  Google Scholar 

  2. F. Albiac and N.J. Kalton, Topics in Banach space theory, Graduate Text in Mathematics 233, Springer, 2006.

    Google Scholar 

  3. D. Alspach, A fixed point free nonexpansive map, Proc. Amer. Math. Soc. 82 (1981), 423–424.

    Article  MATH  MathSciNet  Google Scholar 

  4. R.D. Anderson, Hilbert space is homeomorphic to the countable infinite product of lines, Bull. Amer. Math. Soc. 72 (1966), 515–519.

    Article  MATH  MathSciNet  Google Scholar 

  5. N. Aronszajn and K.T. Smith, Invariant subspaces of completely continuous operators, Ann. Math. 60 (1954), 345–350.

    Article  MathSciNet  Google Scholar 

  6. A. Avilés, Compact space that do not map onto finite products, Fund. Math. 202 (2009), 81–96.

    Article  MATH  MathSciNet  Google Scholar 

  7. S.M. Bates, W.B. Johnson, J. Lindenstrauss, D. Preiss, and G. Schechtman, Affine approximation of Lipschitz functions and nonlinear quotients, GAFA, Geom. Funct. Anal. 9 (1999), 1092–1127.

    Article  MATH  MathSciNet  Google Scholar 

  8. Y. Benyamini and J. Lindenstrauss, Geometric nonlinear functional analysis, Vol. 1, Colloquium Publications 48, American Mathematical Society, 2000.

    Google Scholar 

  9. Y. Benyamini and Y. Sternfeld, Spheres in infinite-dimensional normed spaces are Lipschitz contractible, Proc. Amer. Math. Soc. 88 (1983), 439–445.

    Article  MATH  MathSciNet  Google Scholar 

  10. C. Bessaga, On topological classification of complete linear metric spaces, Fund. Math. 56 (1965), 251–288.

    MATH  MathSciNet  Google Scholar 

  11. C. Bessaga, Topological equivalence of nonseparable reflexive Banach spaces, ordinal resolutions of identity and monotone bases, Ann. Math. Studies 69 (1972), 3–14.

    MathSciNet  Google Scholar 

  12. C. Bessaga and A. Pełczyński, Selected topics in infinite-dimensional topology, Polish Scientific Publishers, Warszawa, 1975.

    MATH  Google Scholar 

  13. R. Cauty, Un espace métrique linéaire qui n’est pas un rétracte absolu, Fund. Math. 146 (1994), 85–99.

    MATH  MathSciNet  Google Scholar 

  14. R. Cauty, Solution du problème de point fixe de Schauder, Fund. Math. 170 (2001), 231–246.

    Article  MATH  MathSciNet  Google Scholar 

  15. M. Cepedello, On regularization in superreflexive Banach spaces by infimal convolution formulas, Studia Math. 129 (1998), 265–284.

    MATH  MathSciNet  Google Scholar 

  16. R. Deville, V.P. Fonf, and P. Hájek, Analytic and C k approximations of norms in separable Banach spaces, Studia Math. 120 (1996), 61–74.

    MATH  MathSciNet  Google Scholar 

  17. R. Deville, V.P. Fonf, and P. Hájek, Analytic and polyhedral approximation of convex bodies in separable polyhedral Banach spaces, Israel J. Math. 105 (1998), 139–154.

    Article  MATH  MathSciNet  Google Scholar 

  18. T. Domínguez-Benavides, A renorming of some nonseparable Banach spaces with the fixed point property, J. Math. Anal. Appl. 350(2) (2009), 525–530.

    Article  MATH  MathSciNet  Google Scholar 

  19. J. Dugundji, Topology, Allyn and Bacon, Inc., Boston, 1966.

    MATH  Google Scholar 

  20. N. Dunford and J.T. Schwartz, Linear operators, Part I, Interscience, New York, 1958.

    Google Scholar 

  21. P. Enflo, J. Lindenstrauss, and G. Pisier, On the three space problem, Math. Scand. 36 (1975), 189–210.

    MathSciNet  Google Scholar 

  22. A. Genel and J. Lindenstrauss, An example concerning fixed points, Israel J. Math. 22 (1975), 81–86.

    Article  MATH  MathSciNet  Google Scholar 

  23. G. Godefroy and N.J. Kalton, Lipschitz-free Banach spaces, Studia Math. 159 (2003), 121–141.

    Article  MATH  MathSciNet  Google Scholar 

  24. G. Godefroy, N.J. Kalton, and G. Lancien, Lipschitz isomorphisms and subspaces of c 0 (ℕ), Geom. Funct. Anal. 10 (2000), 798–820.

    Article  MATH  MathSciNet  Google Scholar 

  25. G. Godefroy, N.J. Kalton, and G. Lancien, Szlenk indices and uniform homeomorphisms, Trans. Amer. Math. Soc. 353 (2001), 3895–3918.

    Article  MATH  MathSciNet  Google Scholar 

  26. K. Goebel, On a fixed point theorem for multivalued nonexpansive mappings, Annal. Univ. Mariae Curie-Skłodowska 29 (1975), 70–72.

    Google Scholar 

  27. E. Gorelik, The uniform non-equivalence of \({{\mathcal{L}}}_p\) and ℓ p , Israel J. Math. 87 (1994), 1–8.

    Article  MATH  MathSciNet  Google Scholar 

  28. S.P. Gul’ko, The space \(C_p(X)\) for countable compact X is uniformly homeomorphic to c 0, Bull. Pol. Acad. Sci. 36 (1988), 391–396.

    MATH  MathSciNet  Google Scholar 

  29. S.P. Gulko, Uniformly homeomorphic spaces of functions, Proc. Steklov Inst. Math. 3 (1993), 87–93.

    MathSciNet  Google Scholar 

  30. P. Hájek, V. Montesinos, J. Vanderwerff, and V. Zizler, Biorthogonal systems in Banach spaces, CMS Books in Mathematics, Canadian Mathematical Society, Springer, 2007.

    Google Scholar 

  31. S. Heinrich and P. Mankiewicz, Applications of ultrapowers to the uniform and Lipschitz classification of Banach spaces, Studia Math. 73 (1982), 225–251.

    MATH  MathSciNet  Google Scholar 

  32. M. Johanis, Approximation of Lipschitz mappings, Serdica Math. 29 (2003), 141–148.

    MATH  MathSciNet  Google Scholar 

  33. M. Johanis and J. Rychtář, On uniformly Gâteaux smooth norms and normal structure, Proc. Amer. Math. Soc. 135 (2007), 1511–1514.

    Article  MATH  MathSciNet  Google Scholar 

  34. W.B. Johnson, J. Lindenstrauss, and G. Schechtman, Banach spaces determined by their uniform structures, Geom. Funct. Anal. 6 (1996), 430–470.

    Article  MATH  MathSciNet  Google Scholar 

  35. W.B. Johnson and M. Zippin, Subspaces and quotient spaces of \((\sum G_n)_{\ell_p}\) and \((\sum G_n)_{0}\), Israel J. Math. 17 (1974), 50–55.

    Article  MATH  MathSciNet  Google Scholar 

  36. M.I. Kadec, Proof of topological equivalence of separable infinite-dimensional Banach spaces, Funct. Anal. Appl. 1 (1967), 53–62.

    Article  MathSciNet  Google Scholar 

  37. N.J. Kalton, Quasi-Banach spaces, Handbook of Banach Spaces II, Editors W.B. Johnson and J.Lindenstrauss, Elsevier, 2003, 1099–1130.

    Google Scholar 

  38. N.J. Kalton, Coarse and uniform embeddings into reflexive spaces, Q. J. Math. 58 (2007), 393–414.

    Article  MATH  MathSciNet  Google Scholar 

  39. N.J. Kalton, The nonlinear geometry of Banach spaces, Rev. Mat. Complutense 21 (2008), 7–60.

    MATH  MathSciNet  Google Scholar 

  40. N.J. Kalton, Lipschitz and uniform embeddings into \(\ell_{\infty}\), to appear.

    Google Scholar 

  41. L.A. Karlovitz, On nonexpansive mappings, Proc. Amer. Math. Soc. 55 (1976), 321–325.

    Article  MATH  MathSciNet  Google Scholar 

  42. O.H. Keller, Die Homoiomorphie der kompakten konvexen Mengen in Hilbertschen Raum, Math. Ann. 105 (1931), 748–758.

    Article  MathSciNet  Google Scholar 

  43. W.A. Kirk, A fixed point theorem for mappings which do not increase distances, Amer. Math. Monthly 72 (1965), 1004–1006.

    Article  MATH  MathSciNet  Google Scholar 

  44. V.L. Klee, Mappings into normed linear spaces, Fund. Math. 49 (1960/1961), 25–34.

    MathSciNet  Google Scholar 

  45. V.L. Klee, Dispersed Chebyshev sets and coverings by balls, Math. Ann. 257 (1981), 251–260.

    Article  MATH  MathSciNet  Google Scholar 

  46. V.L. Klee, Do infinite-dimensional Banach spaces admit nice tilings? Studia Scientiarum Mathematicarum Hungarica 21 (1986), 415–427.

    MathSciNet  Google Scholar 

  47. J.M. Lasry and P.L. Lions, A remark on regularization in Hilbert spaces, Israel J. Math. 55 (1986), 257–266.

    Article  MATH  MathSciNet  Google Scholar 

  48. J. Lindenstrauss, On nonlinear projections in Banach spaces, Michigan Math. J. 11 (1954), 263–287.

    MathSciNet  Google Scholar 

  49. J. Lindenstrauss and L. Tzafriri, Classical Banach spaces I, Sequence spaces, Springer, 1977.

    Google Scholar 

  50. S. Mazur, Une remarque sur l’homémorphic des champs fonctionnels, Studia Math. 1 (1929), 83–85.

    Google Scholar 

  51. J. Milnor, Analytic proofs of the “Hairy Ball Theorem” and the Brouwer fixed point theorem. Ameri. Math. Monthly, 85(7) (1978), 521–524.

    Article  MATH  MathSciNet  Google Scholar 

  52. I. Namioka, Fragmentability in Banach spaces: Interaction of topologies, Lecture Notes Paseky, 1999.

    Google Scholar 

  53. A.M. Nemirovski and S.M. Semenov, On polynomial approximation in function spaces, Mat. Sbornik 21 (1973), 255–277.

    Article  Google Scholar 

  54. J. Pelant, P. Holický, and O. Kalenda, C(K) spaces which cannot be uniformly embedded into \(c_0(\varGamma)\), Fund. Math. 192 (2006), 245–254.

    Article  MATH  MathSciNet  Google Scholar 

  55. D. Preiss, Differentiability of Lipschitz functions on Banach spaces, J. Funct. Anal. 91 (1990), 312–345.

    Article  MATH  MathSciNet  Google Scholar 

  56. M. Raja, Measurabilité de Borel et renormages dans les espaces de Banach, PhD Thesis, Université de Bourdeaux, 1998.

    Google Scholar 

  57. M. Ribe, On uniformly homeomorphic normed spaces, Ark. Math. 14 (1976), 237–244.

    Article  MATH  MathSciNet  Google Scholar 

  58. M. Ribe, Existence of separable uniformly homeomorphic nonisomorphic Banach spaces, Israel J. Math. 48 (1984), 139–147.

    Article  MATH  MathSciNet  Google Scholar 

  59. J. Schauder, Der Fizpunktsatz in Funktionalräumen, Studia Math. 2 (1930), 171–180.

    Google Scholar 

  60. H. Toruńczyk, Characterizing Hilbert space topology, Fund. Math. 111 (1981), 247–262.

    MATH  MathSciNet  Google Scholar 

  61. S. Troyanski, On topological equivalence of spaces \(c_0(\aleph)\) and \(\ell_1(\aleph)\), Bull. Acad. Polon. Sci. Ser. Sci. Math. Astr. et Phys. 15 (1967), 389–396.

    Google Scholar 

  62. I.G. Tsarkov, On global existence of an implicit function, Russian Acad. Sci. Sbornik, 79 (1994), 287–313.

    Article  MathSciNet  Google Scholar 

  63. B. Turett, A dual view of a theorem of Baillon, Lect. Notes Pure Appl. Mat. 80 (1982), Marcel Dekker 1982.

    Google Scholar 

  64. A. Tychonoff, Ein Fixpunktsatz, Math. Annalen 111 (1935), 767–776.

    Article  MathSciNet  Google Scholar 

  65. J. Vanderwerff, Smooth approximations in Banach spaces, Proc. Amer. Math. Soc. 115 (1992), 113–120.

    Article  MATH  MathSciNet  Google Scholar 

  66. D. Werner, A proof of the Markov–Kakutani fixed point theorem via the Hahn–Banach theorem, Extracta Math. 8 (1993), 37–38.

    MATH  MathSciNet  Google Scholar 

  67. V. Zizler, On some rotundity and smoothness properties of Banach spaces, Dissert. Math. 87 (1971), 1–33.

    MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Mathematical Institute of the Academy of Sciences of the Czech Republic, Žitná 25, Praha 1, 11567, Prague, Czech Republic

    Marián Fabian & Petr Hájek

  2. Department of Mathematics, Faculty of Electrical Engineering, Czech Technical University in Prague, Technická 2, 16627, Prague, Czech Republic

    Petr Habala

  3. Departamento de Matematica Aplicada, Universidad Politécnica de Valencia, Camino de Vera s/n, 46022, Valencia, Spain

    Vicente Montesinos

  4. Department of Mathematical and Statistical Sciences, University of Alberta, Central Academic Building, Edmonton, T6G 2G1, Alberta, Canada

    Václav Zizler

Authors
  1. Marián Fabian
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  2. Petr Habala
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  3. Petr Hájek
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  4. Vicente Montesinos
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  5. Václav Zizler
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Correspondence to Marián Fabian .

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Fabian, M., Habala, P., Hájek, P., Montesinos, V., Zizler, V. (2011). Basics in Nonlinear Geometric Analysis. In: Banach Space Theory. CMS Books in Mathematics. Springer, New York, NY. https://doi.org/10.1007/978-1-4419-7515-7_12

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