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

Abstract

Let (Xn‖⋅‖n) denote a sequence of real Banach spaces. LetX=⊕1Xn={(xn):xn∈Xnfor anyn∈N,∑n=1∞‖xn‖n 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.