Every Strongly Remotal Subset In Banach Spaces is a Singleton

Let X be a Banach space, and E ⊂ X be a non-empty closed bounded subset of X. The set E is called proximinal in X if for all x ∈ X there is some e ∈ E such that Çx - eÇ = inf{Çx - yÇ : y ∈ E}. E is called remotal in X if for all x ∈ X, there exists e ∈ E such that Çx - eÇ = sup{Çx - yÇ : y ∈ E}. The concept of strong proximinality is well known by now in the literature, and many results were obtained. In this paper we introduce the concept of strong remotality of sets. Many results are presented.