ON APPROXIMATION PROPERTIES OF l1-TYPE SPACES

Let (X-n || center dot ||(n)) denote a sequence of real Banach spaces. Let X = circle plus X-n = {(x(n)) : x(n) is an element of X-n for any n is an element of N, Sigma(infinity) ||x(n)||(n) <infinity} 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 Nuniq = {x is an element of X : card (P-Y (x)) > 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.