Mackey continuity of convex functions on dual Banach spaces: a review

• A. J. Wrobel ^[1]
  1. [1] 15082 East County Road 600N, Charleston, Illinois, 61920-8026, United States
• Localización: Extracta mathematicae, ISSN-e 0213-8743, Vol. 35, Nº 2, 2020, págs. 185-195
• Idioma: inglés
• DOI: 10.17398/2605-5686.35.2.185
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• Resumen

  • A convex (or concave) real-valued function, f , on a dual Banach space P is continuous for the Mackey topology m (P∗, P ) if (and only if) it is Mackey continuous on bounded subsets of P∗ . Equivalence of Mackey continuity to sequential Mackey continuity follows when P is strongly weakly compactly generated, e.g., when P = L1(T ), where T is a set that carries a sigma-finite measure σ. This result of Delbaen, Orihuela and Owari extends their earlier work on the case that P∗ is either L∞ (T ) or a dual Orlicz space. An earlier result of this kind is recalled also: it derives Mackey continuity from bounded Mackey continuity for a nondecreasing concave function, F , that is defined and finite only on the nonnegative cone L∞+. Applied to a linear f , the Delbaen-Orihuela-Owari result shows that the convex bounded Mackey topology is identical to the Mackey topology, i.e., cbm (P∗, P ) = m (P∗, P ); here, this is shown to follow also from Grothendieck’s Completeness Theorem. As for the bounded Mackey topology, bm (P∗, P ), it is conjectured here not to be a vector topology, or equivalently to be strictly stronger than m (P∗, P ), except when P is reflexive.

• Referencias bibliográficas
  • [1] C.D. Aliprantis, K.C. Border, “ Infinite Dimensional Analysis ”, Springer, Berlin-Heidelberg-New York, 2006.doi.org/10.1007/3-540-29587-9
  • [2] C.D. Aliprantis, O. Burkinshaw, “ Locally Solid Riesz Spaces with Applications to Economics ”, American Mathematical Society, Providence,...
  • [3] T.F. Bewley, Existence of equilibria in economies with infinitely many com- modities, J. Econom. Theory 4 (3) (1972), 514 – 540. doi.org/10.1016/0022-0531(72)90136-6
  • [4] H.S. Collins, Completeness and compactness in linear topological spaces, Trans. Amer. Math. Soc. 79 (1955), 256 – 280. doi.org/10.1090/S0002-9947-1955-0069386-1
  • [5] J.B. Cooper, “ Saks Spaces and Applications to Functional Analysis ”, Second edition, North-Holland, Amsterdam, 1987. doi.org/10.1016/S0304-0208(08)72315-6
  • [6] F. Delbaen, J. Orihuela, Mackey constraints for James’s compactness theorem and risk measures, J. Math. Anal. Appl. 485(1) (2020), Article...
  • [7] F. Delbaen, K. Owari, Convex functions on dual Orlicz spaces, Positivity 23 (5) (2019), 1051 – 1064. doi.org/10.1007/s11117-019-00651-x
  • [8] J. Gil de Lamadrid, Topology of mappings and differentiation processes, Illinois J. Math. 3 (1959), 408 – 420. doi.org/10.1215/ijm/1255455262
  • [9] J. Gómez Gil, On local convexity of bounded weak topologies on Banach spaces, Pacific J. Math. 110 (1) (1984), 71 – 76. doi.org/10.2140/pjm.1984.110.71
  • [10] M. González, J.M. Gutiérrez, The compact weak topology on a Banach space, Proc. Roy. Soc. Edinburgh Sect. A 120 (3-4) (1992), 367 –...
  • [11] A. Horsley, A.J. Wrobel, Efficiency rents of storage plants in peak-load pricing, II: hydroelectricity, LSE, 1999, STICERD DP TE/99/372....
  • [12] A. Horsley, A.J. Wrobel, Localisation of continuity to bounded sets for nonmetrisable vector topologies and its applications to economic...
  • [13]A. Horsley, A.J. Wrobel, Boiteux’s solution to the shifting-peak problem and the equilibrium price density in continuous time, Econom....
  • [14] A. Horsley, A.J. Wrobel, Demand continuity and equilibrium in Banach commodity spaces, in “Game Theory and Mathematical Economics”, Banach...
  • [15] A. Horsley, A.J. Wrobel, “The Short-Run Approach to Long-Run Equilibrium in Competitive Markets: A General Theory with Application to...
  • [16] M. Nowak, On the finest Lebesgue topology on the space of essentially bounded measurable functions, Pac. J. Math. 140 (1) (1989), 151...
  • [17] H.H. Schaefer, “ Topological Vector Spaces ”, Second edition, Springer- Verlag, New York, 1999. doi.org/10.1007/978-1-4612-1468-7
  • [18] G. Schlüchtermann, R.F. Wheeler, On strongly WCG Banach spaces, Math. Z. 199(3) (1988), 387 – 398. doi.org/10.1007/bf01159786
  • [19] A. Wiweger, Linear spaces with mixed topology, Studia Math. 20 (1961), 47 – 68. doi.org/10.4064/sm-20-1-47-68
  • [20] A.J. Wrobel, Bounded topologies on Banach spaces and some of their uses in economic theory: a review, (2020). arXiv:2005.05202