Abstract
This chapter discusses some problems, conjectures, and perspectives, of topological nature in nonlinear and functional analysis. The chapter considers (E, ∥ ·∥ ) to be a real normed space. A nonempty set A ⊂E is said to be antiproximinal with respect to ∥ ·∥ if, for every x ∈E \\ A and every y ∈A, one has ∥x–y ∥ > infz∈A ∥x –z∥. The chapter presents a Conjecture that states “There exists a non-complete real normed space E with the following property: for every nonempty convex set A ⊂ E which is antiproximinal with respect to each norm on E, the interior of the closure of A is nonempty.” The main reason for the study of this conjecture is to give a contribution to open mapping theory in the setting of non-complete normed spaces. Many other problems and theorems are also discussed in the chapter.
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