Nonlinear Analysis 59 (2004) 959 – 971 www.elsevier.com/locate/na Structural properties of some function spaces Yılmaz Yılmaz ∗ Department of Mathematics, Faculty of Art and Science, Inonu University, 44280 Malatya, Turkey Received 22 March 2004; accepted 2 August 2004 Abstract Ferrando and Lüdkovsky (J. Math. Anal. Appl. 274 (2002) 577–585), have investigated some structral properties of the function space c 0 (A,X) for a Hausdorff locally convex space X. In this work, we are mainly interested in the space ℓ(A,X) of all unordered absolute summable functions from a set A into a Hausdorff locally convex space X. The main result of the work is the representation of the elements of ℓ(A,X) and c 0 (A,X). These representations are related with the separability of the spaces and provide us to obtain continuous duals of ℓ(A,X) and c 0 (A,X) for a normed space X. This improves the Ferrando and Lüdkovsky’s investigation with geometric aspects.  2004 Elsevier Ltd. All rights reserved. MSC: primary 46E15; 46E40; 46B20; secondary 46B25; 46B45 Keywords: Representations;Vector-valued function space; Continuous dual; Locally convex space 1. Introduction For a normed space X and a set A, the space ℓ ∞ (A,X) of all bounded X-valued functions x : A → X such that sup{‖x(a)‖ X : a ∈ A} < ∞ has been investigated with several topological and algebraic properties [2–4], where we generally assume that A is an infinite set. In general, ℓ ∞ (A,X) has dual properties with the space X. For example, there are many examples [3,4] of normed spaces of ℓ ∞ (A,X) which ∗ Tel.: +90-422-341-0010X3638; fax: +90-422-3410037. E-mail address: yyilmaz@inonu.edu.tr (Y.Yılmaz). 0362-546X/$ - see front matter  2004 Elsevier Ltd. All rights reserved. doi:10.1016/j.na.2004.08.005