Skip to main content
SpringerLink
Log in

Optimization Without Compactness, and Its Applications

  • Chapter
Operator Theory in Function Spaces and Banach Lattices

Part of the book series: Operator Theory Advances and Applications ((OT,volume 75))

Abstract

In this paper I give a survey of the “near-compactness” properties of norm-bounded convex subsets of Banach Function Spaces which are closed in measure. This originated in the work of G.Ya.Lozanovskiǐ and the author [BL1], and has subsequently been generalized in various directions, including the vector-valued setting. Numerous applications are discussed in the following areas: optimal control, minimax theorems and best approximation in Banach Function Spaces (L 1 being the main example), G.Godefroy’s proof of the weak sequential completeness of L 1/H 10 , geometry of Banach lattices.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 39.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 54.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. K.T. Andrews, J.D. Ward, Proximinality in operator algebras onL1, J. Operator Theory 17 (1987), No.2, 213–221.

    MathSciNet  MATH  Google Scholar 

  2. A. Baernstein, On reflexivity and summability, Studia Math. 42 (1972), 91–94.

    MathSciNet  Google Scholar 

  3. E.J. Balder, New sequential compactness results for spaces of scalarly integrable functions, J. Math. Anal. Appl. 151 (1990), No.1, 1–16.

    Article  MathSciNet  MATH  Google Scholar 

  4. A.A. Basile, A.V. Bukhvalov, M.Ya. Yakubson, The generalized Yosida-Hewitt theorem, Math. Proc. Cambr. Phil. Soc. (submitted).

    Google Scholar 

  5. M. Besbes, Points fixes des contractions définies sur un convexe L0-fermé de L1, C. R. Acad. Sci. Paris, Ser I, 311 (1990), No.5, 243–246.

    MathSciNet  MATH  Google Scholar 

  6. A.V. Bukhvalov, Geometrical properties of Banach spaces of measurable vector-valued functions, Dokl. Akad. Nauk SSSR 239 (1978), No.6, 1279–1282; English transl.: Soviet Math. Dokl. 19 (1978), No.2, 501–505.

    MathSciNet  Google Scholar 

  7. A.V.Bukhvalov, Kernel operators and spaces of measurable vector-valued functions, Dissertation for the degree of doctor of physical and mathematical sciences, LOMI, Leningrad, 1984. (Russian).

    Google Scholar 

  8. A.V. Bukhvalov, Interpolation of linear operators in spaces of vector-valued functions with mixed norm, Sibirsk. Mat. Zh. 28 (1987), No.1, 37–51; English transl.: Siberian Math. J. 28 (1987), 24–36.

    MathSciNet  Google Scholar 

  9. A.V. Bukhvalov, G.Ya. Lozanovskiǐ, On sets closed with respect to convergence in measure in spaces of measurable functions, Dokl. Akad. Nauk SSSR 212 (1973), 1273–1275; English transl.: Soviet Math. Dokl. 14 (1973), 1563–1565.

    MathSciNet  Google Scholar 

  10. A.V. Bukhvalov, G.Ya. Lozanovskiǐ, On sets closed in measure in spaces of measurable functions, Trudy Moskov. Mat. Ob-va 34 (1977), 129–150; English transl.: Trans. Moscow Math. Soc. (1978), issue 2, 127–148.

    MATH  Google Scholar 

  11. A.V. Bukhvalov, G.Ya. Lozanovskiǐ, Representation of linear functionals and operators on vector lattices and certain applications of these representations, in: Theory of Operators in Function Spaces, (Proc. School, Novosibirsk, 1975), “Nauka”, Novosibirsk (1977), 71–98. (Russian).

    Google Scholar 

  12. V. Caselles, Dunford-Pettis operators and the Radon-Nikodym property, Arch. Math. 50 (1988), No.2, 183–188.

    Article  MathSciNet  MATH  Google Scholar 

  13. V. Caselles, A characterization of dual Banach lattices, Proc. Nederl. Akad. Wetensch. A92 (1989), No.1, 35–47.

    MathSciNet  Google Scholar 

  14. C. Castaing, Sur la décomposition de Slaby. Applications aux problèmes de convergence en probabilités, economie mathématique, théorie du controle. Minimisation, Séminaire d’Analyse Convexe, Montpellier 19 (1989), exp. 3.

    Google Scholar 

  15. J. Diestel, J.J. Uhl Jr., Progress in vector measures: 1977–83, Lect. Notes Math. 1033 (1983), 144–192.

    Article  MathSciNet  Google Scholar 

  16. R. Fennich, Application de la methode de relaxation-projection aux problemes min-max, Seminaire d’Analyse Convexe, Montpellier (1980), exp. 8

    Google Scholar 

  17. A. Fougeres, E. Giner, Applications de la décomposition du dual d’un espace d’Orlicz engendré Lϕ: polarité et minimisation “sous compacite”, ϕ-équicontinuité et orthogonalité, C. R. Acad. Sci. Paris A284 (1977), A299–A302.

    MathSciNet  Google Scholar 

  18. A. Fougeres, J.-CI. Peralba, Application au calcul des variations de Voptimisation intégrale convexe, Lecture Notes Math. 1978, 70–99.

    Google Scholar 

  19. D. J.H. Garling, Subsequence principles for vector-valued random variables, Math. Proc. Cambr. Phil. Soc. 86 (1979), No.2, 301–311.

    Article  MathSciNet  MATH  Google Scholar 

  20. T.A. Gillespie, Factorization in Banach function spaces, Proc. Nederl. Akad. Wetensch. A84 (1981), No.3, 287–300.

    MathSciNet  Google Scholar 

  21. E. Giner, Topologies de dualité sur les espaces intégraux de type Orlicz. Applications à I’optimisation. I, II, Séminaire d’Analyse Convexe, Montpellier (1977), exp. 17,18.

    Google Scholar 

  22. E. Giner, Topologies de dualité sur les espaces intégraux de type Orlicz. Applications à I’optimisation, C. R. Acad. Sci. Paris 287 (1978), A425–A428.

    MathSciNet  Google Scholar 

  23. G. Godefroy, Sous-espaces bien disposés deL1 - applications, Trans. Amer. Math. Soc. 286 (1984), 227–249.

    MathSciNet  MATH  Google Scholar 

  24. G. Godefroy, Nicely placed subspaces of Banach lattices, Semesterbericht Funktion-alanalysis, Sommersemester (1984), 205–218.

    Google Scholar 

  25. G. Godefroy, On Riesz subsets of Abelian discrete groups, Israel J. Math. 61 (1988), No.3, 301–331.

    Article  MathSciNet  MATH  Google Scholar 

  26. G. Godefroy, D. Li, Some natural families of M-ideals, Math. Scand. 66 (1990), 249–263.

    MathSciNet  MATH  Google Scholar 

  27. G. Godefroy, F. Lust-Piquard, Some applications of geometry of Banach spaces to harmonic analysis, Colloq. Math. 60/61 (1990), 443–456.

    MathSciNet  Google Scholar 

  28. W. Hensgen, Contributions to the Geometry of Vector-Valued H∞ and L1/H 10 Spaces, Habilitation Thesis, Regensburg, 1992.

    Google Scholar 

  29. S.M. Holland, W.A. Light, L.J. Sulley, On proximinality in L1(T x S), Proc. Amer. Math. Soc. 86 (1982), No.2, 279–282.

    MathSciNet  MATH  Google Scholar 

  30. R.E. Jamison, W.H. Ruckle, Factoring absolutely convergent series, Math. Ann. 224 (1976), 143–148.

    Article  MathSciNet  MATH  Google Scholar 

  31. L.P. Janovskii, Some theorems of nonlinear analysis in non-reflexive Banach spaces, in: Proc. XV-th School on the Theory of Operators in Function Spaces, Part II, Ulianovsk (1990), 136. (Russian).

    Google Scholar 

  32. M.I. Kadec, A. Pełczyński, Bases, lacunary sequences and complemented subspaces in the spaces LP, Studia Math. 21 (1961/62), No.1, 161–176.

    MathSciNet  Google Scholar 

  33. L.V. Kantorovich, G.P. Akilov, Functional Analysis, 2nd rev. ed, “Nauka”, Moscow, 1977 (Russian); English transl.: Pergamon Press, Oxford, 1982.

    Google Scholar 

  34. R. Khalil, Best approximation in LP(X), Math. Proc. Cambr. Phil. Soc. 94 (1983), No.2, 277–279.

    Article  MathSciNet  MATH  Google Scholar 

  35. R. Khalil, Best approximation in L1, Numer. Funct. Anal, and Optim. 9 (1987), No.9–10, 1031–1037.

    Article  MathSciNet  MATH  Google Scholar 

  36. S.V. Kislyakov, More on free interpolation by functions which are regular outside a prescribed set, Zap. Nauchn. Sem. LOMI 107 (1982), 71–88; English transl.: J. Soviet Math. 36 (1987), No.3, 342–352.

    MathSciNet  MATH  Google Scholar 

  37. V.N. Kobzev, On the strong law of large numbers and SX(p, r), MATHTYPEX(p, r)-systems, Soobtch. AN Gruz. SSR (Bull. Acad. Sci. Georgian SSR) 86 (1977), No.1, 53–55. (Russian, English summary).

    MathSciNet  MATH  Google Scholar 

  38. J. Komlós, A generalization of a problem of Steinhaus, Acta Math. Acad. Sci. Hangar. 18 (1967), No.1–2, 217–229.

    Article  MATH  Google Scholar 

  39. C. Lennard, On Komlós and convergence in measure compact, convex subsets of L1 (1990), Preprint.

    Google Scholar 

  40. V.L. Levin, Extremal problems with convex functionals that are lower semicontinuous with respect to convergence in measure, Dokl. Akad. Nauk SSSR 224 (1975), No.6, 1256–1259; English transl.: Soviet Math. Dokl. 16 (1976), No.5, 1384–1388.

    MathSciNet  Google Scholar 

  41. V.L. Levin, Convex Analysis in the Spaces of Measurable Functions, and its Applications in Mathematics and Economics, “Nauka”, Moscow, 1985. (Russian).

    Google Scholar 

  42. D. Li, Espaces L-facteurs de leurs biduaux: bonne disposition, meilleure approximation et propriete de Radon-Nikodym, Quart. J. Math. Oxford (2) 38 (1987), 229–243.

    Article  MATH  Google Scholar 

  43. D. Li, Lifting properties for some quotients of L1 -spaces and other spaces L-summand in their bidual, Math. Z. 199 (1988), 321–329.

    Article  MathSciNet  MATH  Google Scholar 

  44. G.Ya. Lozanovskiǐ, On Banach lattices of Calderón, Dokl. Akad. Nauk SSSR 172 (1967), No.5, 1018–1020; English transl.: Soviet Math. Dokl. 8 (1967).

    MathSciNet  Google Scholar 

  45. G.Ya. Lozanovskiǐ, On some Banach lattices I, Sibirsk. Mat. Zh. 10 (1969), No.3, 584–599; English transl.: Siberian Math. J. 10 (1969), 419–431.

    Google Scholar 

  46. G.Ya. Lozanovskiǐ, Concerning one class of linear operators and its applications to the theory of spaces of measurable functions, Sibirsk. Mat. Zh. 16 (1975), No.4, 755–760; English transl.: Siberian Math. J. 16 (1975), 577–581.

    Google Scholar 

  47. G.Ya. Lozanovskiǐ, Mappings of Banach lattices of measurable functions, Izv. Vyssh. Uchebn. Zaved. Matematika (1978), No.5, 84–86; English transl.: Soviet Math. (Iz. VUZ) 22 (1978), No.5, 61–63.

    Google Scholar 

  48. G.Ya. Lozanovskiǐ, Transformations of ideal Banach spaces by means of concave functions, Qualitative and Approximate Methods of the Investigation of Operator Equations, Jaroslavl’ 3 (1978), 122–147. (Russian).

    Google Scholar 

  49. G.Ya. Lozanovskiǐ, On the conjugate space of a Banach lattice, Theory of Functions, Funct. Anal., and Their Applications (Khar’kov) (1978), No.30, 85–90. (Russian).

    Google Scholar 

  50. D.Q. Luu, A short proof of biting lemma, Seminaire d’Analyse Convexe, Montpellier 19 (1989), exp. 1.

    Google Scholar 

  51. B. Maurey, Théorèmes de factorisation pour les opérateurs linéaires à valeurs dans les espaces Lp, Astérisque (1974), No.11.

    Google Scholar 

  52. P. Nilsson, Interpolation of Banach lattices, Studia Math. 82 (1985), 135–154.

    MathSciNet  MATH  Google Scholar 

  53. G. Pisier, Counterexamples to a conjecture of Grothendieck, Acta Math. 151 (1983), 181–208.

    Article  MathSciNet  MATH  Google Scholar 

  54. G. Pisier, Some applications of the complex interpolation method to Banach lattices, J. D’Analyse Math. 35 (1979), 264–280.

    Article  MathSciNet  MATH  Google Scholar 

  55. S. Reisner, A factorization theorem in Banach lattices and its application to Lorentz spaces, Ann. Inst. Fourier, Grenoble 31 (1981), 239–255.

    Article  MathSciNet  MATH  Google Scholar 

  56. J. Rubio de Francia, Weighted norm inequalities and vector valued inequalities, Lect. Notes Math. 908 (1982), 86–101.

    Article  MathSciNet  Google Scholar 

  57. M. Valadier, Convergence en mesure et optimisation, Travaux du seminaired’analyse convexe, Univ. Languedoc 6 (1976), exp. 14.

    Google Scholar 

  58. K. Yosida, E. Hewitt, Finitely additive measures, Trans. Amer. Math. Soc. 72 (1952), 46–66.

    Article  MathSciNet  MATH  Google Scholar 

  59. A.C. Zaanen, Integration, North-Holland, Amsterdam, 1967.

    MATH  Google Scholar 

  60. A.C. Zaanen, Riesz Spaces. II, North-Holland, Amsterdam, 1983.

    MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, St.Petersburg University of Economics and Finance, Sadovaya street 21, 191023, St.Petersburg, Russia

    Alexander V. Bukhvalov

Authors
  1. Alexander V. Bukhvalov
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

C. B. Huijsmans M. A. Kaashoek W. A. J. Luxemburg B. de Pagter

Additional information

Dedicated to Professor A.C. Zaanen on the occasion of his eightieth birthday

Rights and permissions

Reprints and permissions

Copyright information

© 1995 Birkhäuser Verlag Basel/Switzerland

About this chapter

Cite this chapter

Bukhvalov, A.V. (1995). Optimization Without Compactness, and Its Applications. In: Huijsmans, C.B., Kaashoek, M.A., Luxemburg, W.A.J., de Pagter, B. (eds) Operator Theory in Function Spaces and Banach Lattices. Operator Theory Advances and Applications, vol 75. Birkhäuser Basel. https://doi.org/10.1007/978-3-0348-9076-2_8

Download citation

  • DOI: https://doi.org/10.1007/978-3-0348-9076-2_8

  • Publisher Name: Birkhäuser Basel

  • Print ISBN: 978-3-0348-9896-6

  • Online ISBN: 978-3-0348-9076-2

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics

Search

Navigation