Abstract

We propose a new class of hypertopologies, called here weak∗^* hypertopologies, on the dual space X∗\mathcal{X}^* of a real or complex topological vector space X\mathcal{X}. The most well-studied and well-known hypertopology is the one associated with the Hausdorff metric for closed sets in a complete metric space. Therefore, we study in detail its corresponding weak∗^* hypertopology, constructed from the Hausdorff distance on the field (i.e. R\mathbb{R} or C\mathbb{C}) of the vector space X\mathcal{X} and named here the weak∗^*-Hausdorff hypertopology. It has not been considered so far and we show that it can have very interesting mathematical connections with other mathematical fields, in particular with mathematical logics. We explicitly demonstrate that weak∗^* hypertopologies are very useful and natural structures by using again the weak∗^*-Hausdorff hypertopology in order to study generic convex weak∗^*-compact sets in great generality. We show that convex weak∗^*-compact sets have generically a weak∗^*-dense set of extreme points in infinite dimensions. An extension of the well-known Straszewicz theorem to Gateaux-differentiability (non necessarily Banach) spaces is also proven in the scope of this application.

Author information

Contact details are reproduced from the original publication and may be historical.

Jean-Bernard Bru

Departamento de Matemáticas, Universidad del País Vasco, and: BCAM - Basque Center for Applied Mathematics, Bilbao, Spain
and: IKERBASQUE, Basque Foundation for Science, Bilbao, Spain

Walter de Siqueira Pedra

Departamento de Física Matemática, Universidade de São Paulo, Brazil
and: BCAM - Basque Center for Applied Mathematics, Bilbao, Spain

wpedra@if.usp.br

Keywords

  • Hypertopology
  • Hausdorff distance
  • convex analysis

Mathematics Subject Classification

54B20, 46A03, 52A07, 46A55

References

  1. C. D. Aliprantis, K. C. Border: Infinite Dimensional Analysis. A Hitchhiker’s Guide, 3rd edition, Springer, Berlin (1999).https://doi.org/10.1007/978-3-662-03961-8Google Scholar
  2. E. Asplund: A k-extreme point is the limit of k-exposed points, Israel J. Math. 1/3 (1963) 161–162.https://doi.org/10.1007/BF02759703Google Scholar
  3. E. Asplund: Fréchet differentiability of convex functions, Acta Math. 121 (1968) 31–47.https://doi.org/10.1007/BF02391908Google Scholar
  4. J. Bair: Extension du théorème de Straszewicz, Bull. Soc. Roy. Sci. Liège 45 (1976) 166–168.Google Scholar
  5. J. Bair, R. Fourneau: Etude Géométrique des Espaces Vectoriels. II: Polyèdres et Polytopes Convexes, Springer, Berlin (1980).https://doi.org/10.1007/BFb0089008Google Scholar
  6. P. Bankston: Clopen sets in hyperspaces, Proc. Amer. Math. Soc. 54 (1976) 298–302.https://doi.org/10.1090/S0002-9939-1976-0405332-5Google Scholar
  7. G. Beer: Topologies on Closed and Closed Convex Sets, Mathematics and Its Applications 268, Springer, Dordrecht (1993).https://doi.org/10.1007/978-94-015-8149-3Google Scholar
  8. F. F. Bonsall: The decomposition of continuous linear functionals into non-negative components, Proc. Durham Phil. Soc. 13(A) (1957) 6–11.Google Scholar
  9. O. Brattelli, D. W. Robinson: Operator Algebras and Quantum Statistical Mechanics, Vol. I, 2nd ed., Springer, New York (1996).Google Scholar
  10. J.-B. Bru, W. de Siqueira Pedra: Non-Cooperative Equilibria of Fermi Systems with Long Range Interactions, Memoirs of the American Mathematical Society 224, No. 1052 (2013).https://doi.org/10.1090/S0065-9266-2012-00666-6Google Scholar
  11. J.-B. Bru, W. de Siqueira Pedra: Classical dynamics generated by long-range interactions for lattice fermions and quantum spins, J. Math. Analysis Appl. 493/1 (2021) 124434.https://doi.org/10.1016/j.jmaa.2020.124434Google Scholar
  12. S. N. Burris, H. P. Sankappanavar: A Course in Universal Algebra, Graduate Texts in Mathematics 1, Springer, Berlin (1981).https://doi.org/10.1007/978-1-4613-8130-3_1Google Scholar
  13. S. L. Carvalho, C. R. de Oliveira: Correlation dimension Wonderland theorems, J. Math. Phys. 57 (2016) 063501-1–19.https://doi.org/10.1063/1.4953248Google Scholar
  14. S. L. Carvalho, C. R. de Oliveira: Generic quasilocalized and quasiballistic discrete Schrödinger operators, Proc. Amer. Math. Soc. 144/1 (2016) 129–141.https://doi.org/10.1090/proc/12697Google Scholar
  15. S. L. Carvalho, C. R. de Oliveira: A characterization of singular packing subspaces with an application to limit-periodic operators, Forum Math. 29 (2017) 31–40.https://doi.org/10.1515/forum-2016-0045Google Scholar
  16. S. L. Carvalho, C. R. de Oliveira: Generic zero-Hausdorff and one-packing spectral measures, J. Math. Phys. 62 (2021) 013502.https://doi.org/10.1063/1.5141763Google Scholar
  17. M. Coban, P. S. Kenderov: Dense Gateaux differentiability of the supnorm in C(T) and the topological properties of T, C. R. Acad. Bulgare Sci. 38 (1985) 1603–1604.Google Scholar
  18. J. Dixmier: C ∗ -Algebras, North-Holland, Amsterdam (1977).Google Scholar
  19. M. Fabian: Gâteaux Differentiability of Convex Functions and Topology: Weak Asplund Spaces, Wiley, New York (1997).Google Scholar
  20. V. P. Fonf, J. Lindenstrauss: Some results on infinite-dimensional convexity, Israel J. Math. 108 (1998) 13–32.https://doi.org/10.1007/BF02783039Google Scholar
  21. V. P. Fonf, J. Lindenstrauss, R. R. Phelps: Infinite dimensional convexity, in: Handbook of the Geometry of Banach Spaces, Vol. 1, W. B. Johnson, J. Lindenstrauss (eds.), Chapter 15, Elsevier, Amsterdam (2001) 601–670.Google Scholar
  22. J. Grosberg, M. Krein: Sur la décomposition des fonctionnelles en composantes positives, C. R. (Doklady) Acad. Sci. URSS 25 (1939) 723–726.Google Scholar
  23. R. B. Israel: Convexity in the Theory of Lattice Gases, Princeton Series in Physics, Princeton University Press, Princeton (1979).Google Scholar
  24. J. L. Kelley: General Topology, Graduate Texts in Mathematics 27, Springer, New York (1975).Google Scholar
  25. S. M. Khaleelulla: Counterexamples in Topological Vector Spaces, Lecture Notes in Mathematics 936, Springer, Berlin (1982).https://doi.org/10.1007/BFb0097678Google Scholar
  26. V. L. Klee: Extremal structure of convex sets II, Math. Zeitschrift 69 (1958) 90–104.https://doi.org/10.1007/BF01187394Google Scholar
  27. V. Klee: Some new results on smoothness and rotundity in normed linear spaces, Math. Annalen 139 (1959) 51–63.https://doi.org/10.1007/BF01459822Google Scholar
  28. K. Kuratowski: Topology I, revised and augmented edition, Academic Press, New York (1966).Google Scholar
  29. D. G. Larman, R. R. Phelps: Gateaux differentiability of convex functions on Banach spaces, J. London Math. Soc. s2-20/1 (1979) 115–127.https://doi.org/10.1112/jlms/s2-20.1.115Google Scholar
  30. R. Lucchetti: Convexity and Well-Posed Problems, CMS Books in Mathematics, Springer, New York (2006).https://doi.org/10.1007/0-387-31082-7Google Scholar
  31. S. Mazur: Über konvexe Mengen in linearen normierten Räumen, Studia Math. 4 (1933) 70–84https://doi.org/10.4064/sm-4-1-70-84Google Scholar
  32. W. B. Moors, S. Somasundaram: A Gâteaux differentiability space that is not weak Asplund, Proc. Amer. Math. Soc. 134/9 (2006) 2745–2754.https://doi.org/10.1090/S0002-9939-06-08402-4Google Scholar
  33. R. R. Phelps: Convex Functions, Monotone Operators and Differentiability, 2nd edition, Lecture Notes in Mathematics 1364, Springer, Berlin (1989).https://doi.org/10.1007/978-3-662-21569-2Google Scholar
  34. E. T. Poulsen: A simplex with dense extreme points, Ann. Inst. Fourier (Grenoble) 11 (1961) 83–87.https://doi.org/10.5802/aif.109Google Scholar
  35. W. Rudin: Functional Analysis, McGraw-Hill, New York (1991).Google Scholar
  36. H. H. Schaefer, M. P. Wolff: Topological Vector Spaces, 2nd edition, Springer, New York (1999).https://doi.org/10.1007/978-1-4612-1468-7Google Scholar
  37. B. Sharp: The differentiablility of convex functions on topological linear spaces, Bull. Austral. Math. Soc. 42 (1990) 201–213.https://doi.org/10.1017/S0004972700028379Google Scholar
  38. B. Simon: Operators with singular continuous spectrum. I: General operators, Ann. Math. 141/2 (1995) 131–145.https://doi.org/10.2307/2118629Google Scholar
  39. S. Straszewicz: Über exponierte Punkte abgeschlossener Punktmengen, Fund. Math. 24 (1935) 139–143.https://doi.org/10.4064/fm-24-1-139-143Google Scholar
  40. M. Talagrand: Deux exemples de fonctions convexes, C. R. Acad. Sci. Paris 288 (1979) 461–464.Google Scholar
  41. B. R. Wenner: Finite-dimensional properties of infinite-dimensional spaces, Pacific J. Math. 42/1 (1972) 267–276.https://doi.org/10.2140/pjm.1972.42.267Google Scholar
  42. J. D. Weston: The decomposition of a continuous linear functional into non-negative components, Math. Scand. 5 (1957) 54–56.https://doi.org/10.7146/math.scand.a-10489Google Scholar
  43. V. Zizler: Nonseparable Banach spaces, in: Handbook of the Geometry of Banach Spaces, Vol. 2, Chapter 41, W. B. Johnson and J. Lindenstrauss (eds.), North-Holland, Amsterdam (2003) 1743–1816.https://doi.org/10.1016/S1874-5849(03)80048-7Google Scholar

Suggested citation

J.-B. Bru, W. de Siqueira Pedra. “Weak* Hypertopologies with Application to Genericity of Convex Sets.” Journal of Convex Analysis 29 (2022), No. 1, 13–60. https://doi.org/10.68381/jca29002

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