On approximation properties of l1-type spaces

Abstract

Let $(X_n\Vert \cdot {\Vert }_n)$ denote a sequence of real Banach spaces. Let

$$X={\oplus }_1{X}_n=\left\{\right(x_n):x_n\in X_n\text{for any}n\in \mathbb{N},{\sum }_{n=1}^{\infty }\Vert x_n{\Vert }_n<\infty \}.$$ 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 $\left(X_n\right)$, we characterize finite-dimensional Chebyshev subspaces $Y$ of $X$. Likewise, we show that the set

$$\mathrm{Nuniq}=\{x\in X:card(P_Y\left(x\right))>1\}$$ 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.