J. Math. Anal. Appl. 321 (2006) 193–197 www.elsevier.com/locate/jmaa Some remarks on quasi-Chebyshev subspaces Darapaneni Narayana a , T.S.S.R.K. Rao b,∗ a Department of Mathematics, Indian Institute of Science, Bangalore 560012, India b Stat-Math Unit, Indian Statistical Institute, R. V. College PO, Bangalore 560059, India Received 7 March 2005 Available online 8 September 2005 Submitted by S. Kaijser Abstract We give simple proofs of some results of Mohebi [H. Mohebi, On quasi-Chebyshev subspaces of Banach spaces, J. Approx. Theory 107 (2000) 87–95] on quasi-Chebyshev subspaces.  2005 Published by Elsevier Inc. Keywords: Quasi-Chebyshev subspace; Reflexive subspace 1. Introduction Let X be a real Banach space and let Y ⊂ X be a closed subspace. Consider the set-valued mapping P Y : X → 2 Y defined by P Y (x) =  y ∈ Y : ‖x − y ‖= d(x,Y)  . We say that Y is proximinal in X if P Y (x) =∅ for every x ∈ X. If Y is proximinal, P Y (x) is called the best approximant set of x in Y . We say that Y is Chebyshev if P Y (x) is singleton for every x ∈ X and Y is said to be quasi-Chebyshev if P Y (x) is non-empty and compact for every x ∈ X. We note that Y is Chebyshev (quasi-Chebyshev) if it is Chebyshev (quasi-Chebyshev) in all subspaces of X in which it is contained as a hyperplane, i.e., in every subspace of the form span{Y,x }, x ∈ X. For a Banach space X we denote its closed unit ball by B X and unit sphere by S X . For x ∈ X with d(x,Y) = 1, let Q Y (x) = x − P Y (x). It is easy to see that Q Y (x) ={z ∈ * Corresponding author. E-mail addresses: narayana@math.iisc.ernet.in (D. Narayana), tss@isibang.ac.in (T.S.S.R.K. Rao). 0022-247X/$ – see front matter  2005 Published by Elsevier Inc. doi:10.1016/j.jmaa.2005.08.027