SOME NEW RESULTS ON CERTAIN TYPES OF PROXIMINALITY IN BANACH SPACES

In this paper, we prove that any convex set in a normed space is ε−proximinal. Consequently, every subspace in a Banach space is ε−proximinal. Some other results of proximinality in tensor product spaces are given.


Introduction
Let X be a Banach space and Y be any subset of X.For x ∈ X we define d(x,Y ) = inf y∈Y x − y However, such infimum need not to be attained in Y .If for any x ∈ X there exists some y 0 ∈ Y such that x − y 0 = d(x,Y ), then we say that Y is proximinal in X and y 0 is called a best approximant to x out of Y .Y is called uniquely proximinal if every x ∈ X has a unique best approximant in Y.The problem of whether a set is proximinal or not is a very important problem.
It has many applications in approximation theory in function spaces.In fact one of the most classical open conjecture in approximation theory is : If E is a uniquely proximinal set in a Hilbert space X,then E is convex.We refer to [1], [2], [3], and [10] for many results on proximinality.Many other types of proximinality were introduced over the years.The concept of ε− proximinality was introduced later.Many papers were written on such concept, see [4], [5], [6], [7], [8] and [9] .In this paper we prove that every set in a Banach space is ε−proximinal.Some other results on proximinality in tensor product spaces are presented.
In this section we prove that every set in a Banach space is ε− proximinal.We start with the definition of ε− proximinality.Definition 2.1.Let G be a subset of a Banach space X.Let ε > 0 be given and x ∈ X.Then we However, the converse need not be true.The set then 0 is the best approximation for x in A.
(2) If x ∈ A, then x is the best approximation to itself.
(3) If x ≥ 1, then for any ε > 0 take x 0 ∈ [1 − ε, 1).Then x 0 is an ε-best approximation of x in A. This is true since: Now we prove the main theorem in this section.
Theorem 2.1.Let E be any set in a Banach space X.Then for any ε > 0, E is ε− proximinal in X.
Proof.Let x ∈ X be any element.
This is a contradiction.

Proximinality In Injective Tensor Product Spaces
We recall the following definition Definition 3.1.Let X be a Banach space.Then X is said to have the approximation property if for every compact subset K of X and every ε > 0 there exists a finite rank operator S : X → X such that For the next result, We need the following two Lemmas.
Lemma 3.1.[11], Let X and Y be Banach spaces such that X * has the Radon-Nicodym property and either X * or Y * has the approximation property .Then [10], Let X and Y be Banach spaces such that X * has the approximation property.
Let X be a reflexive space with the approximation property.If H is a finite dimensional subspace of a Banach space Y , then X Proof.Since X is reflexive, then so is X * .Hence, X * has the Radon-Nicodym property; [11].
Also H * has the approximation property, since the Identity operator is a finite rank operator on H * such that Ix − x ≤ ε for every x ∈ H * and every ε > 0 Thus by Lemma 3.1 we have Now, since X is reflexive then X * * = X has the approximation property.Further, any A ∈ L(X * , H * * ) = L(X * , H) is compact.This is because for any bounded subset M ⊆ X * we have A(M) is closed and bounded in H and hence is compact.So, by Lemma 3.2 we get But every reflexive subspace is proximinal [12].Thus X

Proximinality In Projective Tensor Product Spaces
Let X and Y be two Banach spaces, and let X ∧ ⊗ Y denote the completed projective tensor product of X and Y .Then x i y i < ∞, where x i ∈ X and y i ∈ Y ∀i ∈ N}; see [10] Theorem 4.1.Let E and F be two subsets of X and Y respectively.We let [E] and [F] denote the span of the sets E and F respectively.Assume that [E] is separable dual space in X and [F] By the definition of the infimum; there exists a sequence x i ⊗ e i where x i ∈ [E] and {e 1 , e 2 , , , , e n } is a basis for [F]; [10] Thus w m = ∑ n i=1 x m i ⊗ e i , where ).Then each one of them has a w * −convergent subsequence; being a bounded sequence in a separable dual space space [E], (Helly's selection theorem).
We can extract w * −convergent subsequences with uniform index, say (x  Proof.It follows by proceeding as the proof in Theorem 4.1 and using the fact that every bounded sequence in reflexive space has a w− convergent subsequence; [11].

2 ⊗
e 2 + .. + x m j n ⊗ e n Then (u m j ) is a subsequence of (w m ) which is w * −convergent, say to u.Thus we have for any f in the unit ball of the predual space of [E] ∧ ⊗ [F] = (G ∨ ⊗ H) * where G and H are the predual spaces of [E] and [F] respectively; Lemma 3.1.