ABSTRACT
Mathematical analysis is based on the concept of limit. Convergence of sequences, continuity, differentiability and integrability of functions, are all based on the consideration of certain limiting processes. The principal purpose of this chapter is to develop an axiomatic generalisation of Real Analysis to the more abstract realm of metric spaces. A closely associated theme will be the study of a privileged class of metric spaces, called normed spaces. The chapter abstracts well-known concepts of Real Analysis to the metric setting. The idea is to revisit in each case the Euclidean definitions and isolate the essential idea which depends only on the underlying Euclidean distance. It introduces the notion of a bounded set in a metric space. The idea is a direct extension of boundedness stemming from our Euclidean intuition that a set is bounded set if and only if it is contained into some ball.
