ABSTRACT
Around 1920, a young engineering student investigated certain mathematical structures combining topological and algebraic properties. He introduced the notion of magnitude or length of a vector which led to the study of mathematical structures called normed space and a B-space or complete normed space. Subsequently, these spaces became the foundation of functional analysis. This young man is now recognized as Stefan Banach, a distinguished Polish mathematician, and the complete normed space is now known as a Banach space. It will be seen that a normed space is a metric space and a Banach space is a complete metric space. In this chapter, we present some basic results concerning Banach spaces and mappings defined on them. The concepts of normed space, Banach space, bounded and linear operators and their spaces, convex functionals, and duals of a normed space are discussed. To make the concepts clear, a number of examples are given. The material presented here is essential for proper understanding of various branches of mathematics, science and technology.
