## Banach Journal of Mathematical Analysis

### On approximation properties of $l_{1}$-type spaces

#### Abstract

Let $(X_{n}\Vert \cdot \Vert _{n})$ denote a sequence of real Banach spaces. Let

$$X=\bigoplus_{1}X_{n}=\{(x_{n}):x_{n}\in X_{n}\hbox{ for any }n\in \mathbb{N},\sum_{n=1}^{\infty}\Vert x_{n}\Vert _{n}\lt \infty\}.$$ 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 $(X_{n})$, we characterize finite-dimensional Chebyshev subspaces $Y$ of $X$. Likewise, we show that the set

$$\mathrm{Nuniq}=\{x\in X:\operatorname{card}(P_{Y}(x))\gt 1\}$$ is nowhere dense in $Y$, where $P_{Y}$ 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
Accepted: 1 March 2018
First available in Project Euclid: 10 July 2018

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

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

Mathematical Reviews number (MathSciNet)
MR3858755

Zentralblatt MATH identifier
06946297

#### 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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