A known, and easy to establish, fact in Best Approximation Theory is that, if the unit ball of a subspace G of a Banach space X is proximinal in X, then G itself is proximinal in X. We are concerned in this article with the reverse implication, as the knowledge of whether the unit ball is proximinal or not is useful in obtaining information about other problems. We show, by constructing a count...

Let (E, ‖·‖E) be a symmetric space and let Y ⊂ X be a nonempty subset. For x ∈ X denote PY (x) = {y ∈ Y : ‖x− y‖ = dist(x, Y )}. Any element y ∈ PY (x) is called a best approximant in Y to x. A nonempty set Y ⊂ X is called proximinal or set of existence if PY (x) 6= ∅ for any x ∈ X. A nonempty set Y is said to be a Chebyshev set if it is proximinal and PY (x) is a singleton for any x ∈ E. A sym...

Let X be a Banach space, C a closed subset of X , and T : C -+ C a nonexpansive mapping. Conditions are given which assure that if the fixed point set F ( T ) of T has nonempty interior then the Picard iterates of the mapping T always converge to a point of F ( T ) . If T is asymptotically regular, it suffices to assume that the closed subsets of X are densely proximinal and that nested spheres...

Let Y be a proximinal subspace of finite codimension of c0. We show that Y is proximinal in ∞ and the metric projection from ∞ onto Y is Hausdorff metric continuous. In particular, this implies that the metric projection from ∞ onto Y is both lower Hausdorff semi-continuous and upper Hausdorff semi-continuous.

Let X be a normed linear space. We will consider only normed linear spaces over R (Real line), though many of the results we describe hold good for n.l. spaces over C (the complex plane). The dual of X, the class of all bounded, linear functionals on X, is denoted by X∗. The closed unit ball of X is denoted by BX and the unit sphere, by SX . That is, BX = {x ∈ X : ‖x‖ ≤ 1} and SX = {x ∈ X : ‖x‖...

Let G be a reflexive subspace of the Banach space E and let L(I, E) denote the space of all p-Bochner integrable functions on the interval I=[0, 1] with values in E, 1 [ pO.. Given any norm N(· , · ) on R, N nondecreasing in each coordinate on the set R +, we prove that L (I, G) is N-simultaneously proximinal in L(I, E). Other results are also obtained. © 2002 Elsevier Science (USA)