ABSTRACT
Linearity is one of the most sought after properties in mathematics. It is also a profoundly simplifying conceptual tool in the study of mathematical entities from the geometric, algebraic, and analytic points of view. This chapter discusses some elementary properties of linear spaces. The norm of an element of a linear space corresponds to the notion of the length of a vector in two- or three-dimensional vector spaces. The norm may also be interpreted as the distance of any given element of a linear space from the zero element. Indeed, one may introduce the idea of distance between any two elements of a linear space through the norm. This is extremely important for the study of the analytic properties of functionals and operators on a linear space. To understand and exploit the idea of distance as an analytic tool, one will do well to study point spaces with distance defined between all pairs of points before focusing on linear spaces.
