Applied Mathematical Sciences, Vol. 2, 2008, no. 16, 763 - 766 The Nearest Points in Normed Linear Spaces H. Mazaheri Department of Mathematics University of Yazd, Yazd, Iran hmazaheri@yazduni.ac.ir Abstract The purpose of this paper is to introduce and discuss the concept of best approximation and best coapproximation in normed linear spaces. We will generalize some previous results about closed convex subsets to closed subsets. Mathematics Subject Classification: 46B20, 41A50 Keywords: Normed spaces, Proximinal set, Chebyshev set, Best coap- proximation 1. Introduction We know that a point g 0 ∈ M is said to be a best approximation (resp. best coapproximation) for x ∈ X if and only if x − g 0  = x + M  = dist(x, M )(resp. g 0 − g ≤x − g ∀ g ∈ M ). It can be easily proved that g 0 is a best approximation (resp. best coapproximation) for x ∈ X if and only if x − g 0 ∈ ˆ M (resp. x − g 0 ∈ ˘ M ). The set of all best approximations (resp. best coapproximations) of x ∈ X in M is shown by P M (x)(resp. R M (x)). In other words, P M (x)= {g 0 ∈ M : x − g 0 ∈ ˆ M } and R M (x)= {g 0 ∈ M : x − g 0 ∈ ˘ M }. If P M (x)(resp. R M (x)) is non-empty for every x ∈ X , then M is called an Proximinal (resp. coproximinal) set. The set M is Chebyshev (resp. cocheby- shev) if P M (x)(resp. R M (x)) is a singleton set for every x ∈ X . (see [2-6])