Author	Guzman, Alberto. author
Title	Continuous Functions of Vector Variables [electronic resource] / by Alberto Guzman
Imprint	Boston, MA : Birkhรคuser Boston : Imprint: Birkhรคuser, 2002
Connect to	http://dx.doi.org/10.1007/978-1-4612-0083-3
Descript	X, 210 p. 16 illus. online resource

This text is appropriate for a one-semester course in what is usually called adยญ vanced calculus of several variables. The focus is on expanding the concept of continuity; specifically, we establish theorems related to extreme and intermediate values, generalizing the important results regarding continuous functions of one real variable. We begin by considering the function f(x, y, ... ) of multiple variables as a function of the single vector variable (x, y, ... ). It turns out that most of the n treatment does not need to be limited to the finite-dimensional spaces R , so we will often place ourselves in an arbitrary vector space equipped with the right tools of measurement. We then proceed much as one does with functions on R. First we give an algebraic and metric structure to the set of vectors. We then define limits, leading to the concept of continuity and to properties of continuous functions. Finally, we enlarge upon some topological concepts that surface along the way. A thorough understanding of single-variable calculus is a fundamental requireยญ ment. The student should be familiar with the axioms of the real number system and be able to use them to develop elementary calculus, that is, to define continuous junction, derivative, and integral, and to prove their most important elementary properties. Familiarity with these properties is a must. To help the reader, we provide references for the needed theorems

1 Euclidean Space -- 1.1 Multiple Variables -- 1.2 Points and Lines in a Vector Space -- 1.3 Inner Products and the Geometry of Rn -- 1.4 Norms and the Definition of Euclidean Space -- 1.5 Metrics -- 1.6 Infinite-Dimensional Spaces -- 2 Sequences in Normed Spaces -- 2.1 Neighborhoods in a Normed Space -- 2.2 Sequences and Convergence -- 2.3 Convergence in Euclidean Space -- 2.4 Convergence in an Infinite-Dimensional Space -- 3 Limits and Continuity in Normed Spaces -- 3.1 Vector-Valued Functions in Euclidean Space -- 3.2 Limits of Functions in Normed Spaces -- 3.3 Finite Limits -- 3.4 Continuity -- 3.5 Continuity in Infinite-Dimensional Spaces -- 4 Characteristics of Continuous Functions -- 4.1 Continuous Functions on Boxes in Euclidean Space -- 4.2 Continuous Functions on Bounded Closed Subsets of Euclidean Space -- 4.3 Extreme Values and Sequentially Compact Sets -- 4.4 Continuous Functions and Open Sets -- 4.5 Continuous Functions on Connected Sets -- 4.6 Finite-Dimensional Subspaces of Normed Linear Spaces -- 5 Topology in Normed Spaces -- 5.1 Connected Sets -- 5.2 Open Sets -- 5.3 Closed Sets -- 5.4 Interior, Boundary, and Closure -- 5.5 Compact Sets -- 5.6 Compactness in Infinite Dimensions -- Solutions to Exercises -- References