Banach Journal of Mathematical Analysis

On approximation properties of l1-type spaces

Maciej Ciesielski and Grzegorz Lewicki

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Abstract

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

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

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

Permanent link to this document
https://projecteuclid.org/euclid.bjma/1531209675

Digital Object Identifier
doi:10.1215/17358787-2018-0005

Mathematical Reviews number (MathSciNet)
MR3858755

Zentralblatt MATH identifier
06946297

Subjects
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

Keywords
Banach spaces continuous selection for the metric projection Chebyshev subspaces

Citation

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. https://projecteuclid.org/euclid.bjma/1531209675


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