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Strong proximinality and renormings

Author: Darapaneni Narayana
Journal: Proc. Amer. Math. Soc. 134 (2006), 1167-1172
MSC (2000): Primary 46B20
Published electronically: October 25, 2005
MathSciNet review: 2196053
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Abstract: We characterize finite-dimensional normed linear spaces as strongly proximinal subspaces in all their superspaces. A connection between upper Hausdorff semi-continuity of metric projection and finite dimensionality of subspace is given.

References [Enhancements On Off] (What's this?)

  • 1. Frank Deutsch, Walter Pollul, and Ivan Singer, On set-valued metric projections, Hahn-Banach extension maps, and spherical image maps, Duke Math. J. 40 (1973), 355–370. MR 0313759
  • 2. Robert Deville, Gilles Godefroy, and Václav Zizler, Smoothness and renormings in Banach spaces, Pitman Monographs and Surveys in Pure and Applied Mathematics, vol. 64, Longman Scientific & Technical, Harlow; copublished in the United States with John Wiley & Sons, Inc., New York, 1993. MR 1211634
  • 3. Carlo Franchetti and Rafael Payá, Banach spaces with strongly subdifferentiable norm, Boll. Un. Mat. Ital. B (7) 7 (1993), no. 1, 45–70 (English, with Italian summary). MR 1216708
  • 4. G. Godefroy, V. Indumathi, and F. Lust-Piquard, Strong subdifferentiability of convex functionals and proximinality, J. Approx. Theory 116 (2002), no. 2, 397–415. MR 1911087,
  • 5. G. Godefroy and V. Indumathi, Strong proximinality and polyhedral spaces, Rev. Mat. Complut. 14 (2001), no. 1, 105–125. MR 1851725,
  • 6. A. Pełczyński, All separable Banach spaces admit for every 𝜖>0 fundamental total and bounded by 1+𝜖 biorthogonal sequences, Studia Math. 55 (1976), no. 3, 295–304. MR 0425587
  • 7. W. Pollul, Topologien auf Mengen von Teilmengen und Stetigkeit von mengenwertigen metrischen Projektionen, Diplomarbeit, Bonn, 1967.
  • 8. Walter Pollul, Reflexivität und Existenz-Teilräume in der linearen Approximationstheorie, Gesellschaft für Mathematik und Datenverarbeitung, Bonn, 1972. Gesellschaft für Mathematik und Datenverarbeitung, Bonn, Ber. No. 53. MR 0445193
  • 9. Ivan Singer, On normed linear spaces which are proximinal in every superspace, J. Approximation Theory 7 (1973), 399–402. MR 0346397
  • 10. Ivan Singer, Best approximation in normed linear spaces by elements of linear subspaces, Translated from the Romanian by Radu Georgescu. Die Grundlehren der mathematischen Wissenschaften, Band 171, Publishing House of the Academy of the Socialist Republic of Romania, Bucharest; Springer-Verlag, New York-Berlin, 1970. MR 0270044

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Additional Information

Darapaneni Narayana
Affiliation: Stat-Math Unit, Indian Statistical Institute, R. V. College P.O., Bangalore 560059, India
Address at time of publication: Department of Mathematics, Indian Institute of Science, Bangalore 560012, India

Keywords: Strongly proximinal subspace, reflexive space, finite-dimensional space.
Received by editor(s): August 2, 2004
Received by editor(s) in revised form: November 11, 2004
Published electronically: October 25, 2005
Communicated by: Jonathan M. Borwein
Article copyright: © Copyright 2005 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.