Linear spaces

Abstract

The axioms of a linear or vector space have been chosen to display some of the algebraic properties common to many classes of functions appearing frequently in analysis. Of these examples there is no doubt that the most fundamental, and earliest, examples are furnished by the n-dimensional Euclidean spaces and their vector algebras. Nearly as important, and the basic examples for most of this book, are many function spaces; for example, C [0, 1], the space of real-valued continuous functions on the closed unit interval, BV [0, 1], the space of functions of bounded variation on the same interval, L ^p [0, 1], the space of those Lebesgue measurable functions on the same interval which have summable p ^th powers, and A (D), the space of all complex-valued functions analytic in a domain D of the complex plane.