Today, I will discuss this research paper by Javed Ali, Professor of Topology and Analysis, BITS Pilani.
What exactly is a proximinal set? It is the set of elements in which for any , you can find the nearest point(s) to it in . More formally, for each such that .
This article says a Banach space is a complete vector space with a norm. One might find the difference between a complete metric space and a complete vector space to be minimal. However, the difference that is crucial here is that not every metric space is a vector space. For example, let be a metric space, satisfying the relevant axioms. However, for , not being defined is possible. However, if is a vector space, then $x+y\in X$. Hence, every normed vector space is a metric space if one were to define , but the converse is not necessarily true.
What is a convex set? This article says a convex set is one in which a line joining any two points in the set lies entirely inside the set. But how can a set containing points contain a line? Essentially, the the convex property implies that every point the line passes through is contained within the convex space. Convexity introduces a geometrical flavor to Banach spaces. It is difficult to imagine what such a line segment would be in the Banach space of matrices (with the norm .
What is a uniformly convex set? This paper says that a uniform convex space is defined thus: such that for and . Multiplying by on both sides, we get . What does this actually mean? The first condition implies that and cannot lie in the same direction. Hence, . As a result, we get , or . As , and as is bounded, can be the lower bound of .
But what is uniform about this condition? It is the fact that does not change with the unit vector being considered, and depends only on .
Now we go on to prove that every closed convex set of every uniformly convex Banach space is proximinal.