ABSTRACT
There are five basic results which, along with their consequences, have played an important role in such diverse areas of mathematical sciences as approximation theory, Fourier analysis, numerical analysis, control theory, optimization, mechanics, mathematical economics, and ordinary and partial differential equations. These results are known as Hahn-Banach Theorem, Banach-Alaoglu Theorem, Uniform Boundedness Principle, Open Mapping and Closed Graph Theorems. Topological forms of the HahnBanach Theorem guarantee the preservation of properties when a functional defined on a subspace is extended over the entire space. It has been established beyond doubt that the existence and uniqueness of solutions of problems in different areas of science and technology are closely related with the topology and related convergence of the underlying space. We discuss here strong, weak and weak* topologies and related convergence. More recent concepts of convergence like convergence of convex sets [15], Г convergence [1,4] and 2-scale convergence [8] have found interesting applications. The Uniform Bounded Principle broadly shows that uniform boundedness of a sequence of bounded linear operators is guaranteed by the boundedness of the set of images. A converse of this result, often called the Banach-Steinhaus Theorem, is given as a solved example. It states that uniform boundedness of a sequence of bounded linear operators, along with pointwise convergence on a dense subset, is sufficient to ensure the pointwise convergence of the sequence on the entire space. The Open Mapping Theorem or the Banach Theorem states that a one-to-one continuous operator from one Banach space onto another has a continuous inverse, that is, the operator maps open sets into open sets. The Closed Graph Theorem demonstrates the continuity of a linear operator with a closed graph. In this chapter, we discuss the above mentioned results and their consequences. References for further reading are given at the end of the chapter.
