ABSTRACT
Functional analysis is the study of various types of vector spaces which are also topological spaces and the linear operators defined on these spaces. As such, it is really a generalization of linear algebra and calculus. The vector spaces which are of interest in this subject include the usual spaces ℝ n and ℂ n but also many which are infinite dimensional such as the space C(X;ℝ n ) discussed in Chapter 2 in which we think of a function as a point or a vector. When the topology comes from a norm, the vector space is called a normed linear space and this is the case of interest here. A normed linear space is called real if the field of scalars is ℝ and complex if the field of scalars is ℂ. We will assume a linear space is complex unless stated otherwise. A normed linear space may be considered as a vector space if we define d(x, y) = ‖x − y‖. As usual, if every Cauchy sequence converges, the metric space is called complete.
