Banach Journal of Mathematical Analysis

On approximation properties of l1-type spaces

Maciej Ciesielski and Grzegorz Lewicki

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Let (Xnn) denote a sequence of real Banach spaces. Let

X=1Xn={(xn):xnXnfor anynN,n=1xnn<}. In this article, we investigate some properties of best approximation operators associated with finite-dimensional subspaces of X. In particular, under a number of additional assumptions on (Xn), we characterize finite-dimensional Chebyshev subspaces Y of X. Likewise, we show that the set

Nuniq={xX:card(PY(x))>1} is nowhere dense in Y, where PY denotes the best approximation operator onto Y. Finally, we demonstrate various (mainly negative) results on the existence of continuous selection for metric projection and we provide examples illustrating possible applications of our results.

Article information

Banach J. Math. Anal., Volume 12, Number 4 (2018), 935-954.

Received: 13 August 2017
Accepted: 1 March 2018
First available in Project Euclid: 10 July 2018

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 41A65: Abstract approximation theory (approximation in normed linear spaces and other abstract spaces)
Secondary: 41A50: Best approximation, Chebyshev systems 46B20: Geometry and structure of normed linear spaces

Banach spaces continuous selection for the metric projection Chebyshev subspaces


Ciesielski, Maciej; Lewicki, Grzegorz. On approximation properties of $l_{1}$ -type spaces. Banach J. Math. Anal. 12 (2018), no. 4, 935--954. doi:10.1215/17358787-2018-0005.

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  • [1] J. Blatter, P. D. Morris, and D. E. Wulbert, Continuity of the set-valued metric projection, Math. Ann. 178 (1968), 12–24.
  • [2] A. L. Brown, A rotund reflexive space having a subspace of codimension two with a discontinuous metric projection, Michigan Math. J. 21 (1974), 145–151.
  • [3] A. L. Brown, Set valued mappings, continuous selections and metric projections, J. Approx. Theory 57 (1989), no. 1, 48–68.
  • [4] A. L. Brown, Metric projections in spaces of integrable functions, J. Approx. Theory 81 (1995), no. 1, 78–103.
  • [5] A. L. Brown, Continuous selections for metric projections in spaces of continuous functions and a disjoint leaves condition, J. Approx. Theory 141 (2006), no. 1, 29–62.
  • [6] A. L. Brown, On lower semi-continuous metric projections onto finite dimensional subspaces of spaces of continuous functions, J. Approx. Theory 166 (2013), 85–105.
  • [7] A. L. Brown, F. Deutsch, V. Indumathi, and P. S. Kenderov, Lower semicontinuity concepts, continuous selections, and set valued metric projections, J. Approx. Theory 115 (2002), no. 1, 120–143.
  • [8] F. R. Deutsch, V. Indumathi, and K. Schnatz, Lower semicontinuity, almost lower semicontinuity, and continuous selections for set-valued mappings, J. Approx. Theory 53 (1988), no. 3, 266–294.
  • [9] F. R. Deutsch and P. H. Maserick, Applications of the Hahn–Banach theorem in approximation theory, SIAM Rev. 9 (1967), 516–530.
  • [10] T. Fischer, A continuity condition for the existence of a continuous selection for a set-valued mapping, J. Approx. Theory 49 (1987), no. 4, 340–345.
  • [11] A. J. Lazar, Spaces of affine continuous function on simplexes, Trans. Amer. Math. Soc. 134 (1968), 503–525.
  • [12] A. J. Lazar, D. E. Wulbert, and P. D. Morris, Continuous selections for metric projections, J. Funct. Anal. 3 (1969), 193–216.
  • [13] W. Li, “Various continuities of metric projections in $L_{1}(T\mu)$” in Progress in Approximation Theory, Academic Press, Boston, 1991, 583–607.
  • [14] R. R. Phelps, Chebyshev subspaces of finite dimension in $L_{1}$, Proc. Amer. Math. Soc. 17 (1966), 646–652.
  • [15] I. Singer, Best Approximation in Normed Spaces by Elements of Linear Subspaces, Grundlehren Math. Wiss. 171, Springer, New York, 1970.
  • [16] I. Singer, The Theory of Best Approximation and Functional Analysis, CBMS-NSF Regional Conf. Ser. in Appl. Math. 13, SIAM, Philadelphia, 1974.