Hostname: page-component-848d4c4894-75dct Total loading time: 0 Render date: 2024-05-10T12:37:37.323Z Has data issue: false hasContentIssue false
  • English
  • Français

Best approximation and intersections of balls in Banach spaces

Published online by Cambridge University Press:  17 April 2009

David T. Yost
David T. Yost
Affiliation:
Department of Mathematics, University of Edinburgh, Edinburgh, Scotland.
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

Let E be a Banach space, M a closed subspace of E with the 3-ball property. It is known that M is proximinal in E, and that its metric projection admits a continuous selection. This means that there is a continuous (generally non-linear) map π: E → M satisfying ‖x−π(x)‖ = d(x, M) for all x in E. Here it is shown that the same conclusion holds under a much weaker hypothesis on M, which we call the 1½-ball property. We also establish that if M has the 1½-ball property in E, then there is a continuous Hahn-Banach extension map from M* to E*.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 20 , Issue 2 , April 1979 , pp. 285 - 300
Copyright
Copyright © Australian Mathematical Society 1979

References

[1]Alfsen, Erik M. and Effros, Edward G., “Structure in real Banach spaces. Part I”, Ann. of Math. (2) 96 (1972), 98–128.CrossRefGoogle Scholar
[2]Ando, T., “Closed range theorems for convex sets and linear liftings”, Pacific J. Math. 44 (1973), 393–410.CrossRefGoogle Scholar
[3]Blatter, Jörg, Grothendieck spaces in approximation theory (Memoirs of the American Mathematical Society, 120. American Mathematical Society, Providence, Rhode Island, 1972).Google Scholar
[4]Bonsall, F.F., Duncan, J., Complete normed algebras (Ergebnisse der Mathematik und ihrer Grenzgebiete, 80. Springer-Verlag, Berlin, Heidelberg, New York, 1973).CrossRefGoogle Scholar
[5]Hennefeld, Julien, “A decomposition for B(X)* and unique Hahn-Banach extensions”, Pacific J. Math. 46 (1973), 197–199.CrossRefGoogle Scholar
[6]Holmes, Richard, Scranton, Bruce, and Ward, Joseph, “Approximation from the space of compact operators and other M-ideals”, Duke Math. J. 42 (1975), 259–269.CrossRefGoogle Scholar
[7]Lima, Åsvald, “Intersection properties of balls and subspaces in Banach spaces”, Trans. Amer. Math. Soc. 227 (1977), 1–62.CrossRefGoogle Scholar
[8]Lindenstrauss, Joram, Extension of compact operators (Memoirs of the American Mathematical Society, 48. American Mathematical Society, Providence, Rhode Island, 1964).Google Scholar
[9]Lindenstrauss, Joram, “On nonlinear projections in Banach spaces”, Michigan Math. J. 11 (1964), 263–287.CrossRefGoogle Scholar
[10]Lindenstrauss, Joram, Tzafriri, Lior, Classical Banach spaces I. Sequence spaces (Ergebnisse der Mathematik und ihrer Grenzgebiete, 92. Springer-Verlag, Berlin, Heidelberg, New York, 1977).Google Scholar
[11]Michael, Ernest, “Continuous selections. I”, Ann. of Math. (2) 63 (1956), 361–382.CrossRefGoogle Scholar
[12]Semadeni, Zbigniew, Banach spaces of continuous functions (instytut Matematyczny Polskiej Akademii Nauk Monografie Matematyczne, 55. Polish Scientific Publishers, Warszawa, 1971).Google Scholar
[13]Smith, R.R. and Ward, J.D., “M-ideal structure in Banach algebras”, J. Funct. Anal. 27 (1978), 337–349.CrossRefGoogle Scholar