Positivity DOI 10.1007/s11117-009-0030-7 Positivity Ball remotality of M -ideals in some function spaces and function algebras Pradipta Bandyopadhyay · Tanmoy Paul · Ashoke K. Roy Received: 15 July 2009 / Accepted: 17 July 2009 © Birkhäuser Verlag Basel/Switzerland 2009 Abstract In this paper, we show that in many function spaces and function algebras, the unit ball of any M -ideal is densely remotal. Keywords Farthest points · Function spaces · DBR subspaces · M -ideals Mathematics Subject Classification (2000) Primary 46B20 · 46E15 1 Introduction We work with complex scalars. The closed unit ball and the unit sphere of a Banach space X are denoted by B X and S X , respectively. For a closed and bounded set M ⊆ X , the farthest distance map φ M is defined as φ M (x ) = sup{‖z - x ‖: z ∈ M }, x ∈ X . For x ∈ X , let F M (x ) ={z ∈ M :‖z - x ‖= φ M (x )}, i.e., the set of points of M farthest from x . Note that this set may be empty. Let R( M ) ={x ∈ X : F M (x )  = ∅}. The following notion has been studied recently in [3, 4]. Definition 1.1 A closed subspace Y of a Banach space X is said to be a densely ball remotal (DBR) subspace of X if R( B Y ) is norm dense in X . P. Bandyopadhyay (B ) · T. Paul Stat-Math Division, Indian Statistical Institute, 203, B.T. Road, Kolkata 700108, India e-mail: pradipta@isical.ac.in T. Paul e-mail: tanmoy_r@isical.ac.in A. K. Roy Department of Mathematics, Ramakrishna Mission Vivekananda University, Belur Math, Howrah 711202, West Bengal, India e-mail: ashokeroy98@yahoo.com