Illinois Journal of Mathematics

Descriptive theory of nearest points in Banach spaces

Robert Kaufman

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Abstract

Let $X$ be a separable Banach space, $Y$ a closed, nonreflexive, linear subspace, and $P$ the set of points admitting a nearest approximation in $Y$. Then $P$ is an analytic set, and has three obvious algebraic properties. By adjusting the norm of $X$, any analytic set of this kind can be realized as the set of elements proximal to $Y$.

Article information

Source
Illinois J. Math., Volume 54, Number 3 (2010), 1157-1162.

Dates
First available in Project Euclid: 3 May 2012

Permanent link to this document
https://projecteuclid.org/euclid.ijm/1336049988

Digital Object Identifier
doi:10.1215/ijm/1336049988

Mathematical Reviews number (MathSciNet)
MR2928349

Zentralblatt MATH identifier
1264.46009

Subjects
Primary: 46B20: Geometry and structure of normed linear spaces 28A05: Classes of sets (Borel fields, $\sigma$-rings, etc.), measurable sets, Suslin sets, analytic sets [See also 03E15, 26A21, 54H05]
Secondary: 46B03: Isomorphic theory (including renorming) of Banach spaces

Citation

Kaufman, Robert. Descriptive theory of nearest points in Banach spaces. Illinois J. Math. 54 (2010), no. 3, 1157--1162. doi:10.1215/ijm/1336049988. https://projecteuclid.org/euclid.ijm/1336049988


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References

  • I. Aharoni and J. Lindenstrauss, Uniform equivalence between Banach spaces, Bull. Amer. Math. Soc. 84 (1978), 281–283.
  • I. Aharoni and J. Lindenstrauss, An extension of a result of Ribe, Israel J. Math. 52 (1985), 59–64.
  • R. G. Bartle and L. M. Graves, Mappings between function spaces, Trans. Amer. Math. Soc. 72 (1952), 400–413.
  • J. Bourgain, $l^{\infty}/c_{0}$ has no equivalent strictly convex norm, Proc. Amer. Math. Soc. 78 (1980), 225–226.
  • M. M. Day, Strict convexity and smoothness of normed spaces, Trans. Amer. Math. Soc. 78 (1955), 516–528.
  • R. Deville, G. Godefroy and V. Zizler, Smoothness and renormings in Banach spaces, Pitman Monographs and Surveys in Pure and Applied Mathematics, vol. 64, Longman Scientific & Technical, Harlow, 1993.
  • J. Dugundji, Topology, Allyn and Bacon Inc., Boston, MA, 1978. Reprinting of the 1966 original, Allyn and Bacon Series in Advanced Mathematics.
  • G. Godefroy and N. J. Kalton, Lipschitz-free Banach spaces, Studia Math. 159 (2003), 121–141. Dedicated to Professor Aleksander Pełczyński on the occasion of his 70th birthday.
  • J. E. Jayne and C. A. Rogers, Selectors, Princeton Univ. Press, Princeton, NJ, 2002.
  • K. Kuratowski, Topology. Vol. I. Academic Press, New York, 1966. New edition, revised and augmented. Translated from the French by J. Jaworowski.
  • J. Lindenstrauss, On nonlinear projections in Banach spaces, Michigan Math. J. 11 (1964), 263–287.
  • J. R. Partington, Equivalent norms on spaces of bounded functions, Israel J. Math. 35 (1980), 205–209.
  • A. Pełczyński, A note on the paper of I. Singer “Basic sequences and reflexivity of Banach spaces”, Studia Math. 21 (1961/1962), 371–374.