This text offers a synthesis of theory and application related to modern techniques of differentiation. Based on undergraduate courses in advanced calculus, the treatment covers a wide range of topics, from soft functional analysis and finite-dimensional linear algebra to differential equations on submanifolds of Euclidean space. Suitable for advanced undergraduate courses in pure and applied mathematics, it forms the basis for graduate-level courses in advanced calculus and differential manifolds. Starting with a brief resume of prerequisites, including elementary linear algebra and point set topology, the self-contained approach examines liner algebra and normed vector spaces, differentiation and calculus on vector spaces, and the inverse- and implicit-function theorems. A final chapter is dedicated to a consolidation of the theory as stated in previous chapters, in addition to an introduction to differential manifolds and differential equations.
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In this chapter K will always denote either the field of real numbers, R, or the field of complex numbers, C. | | will denote the absolute value or modulus of numbers in K.
Let E be a vector space over K of not necessarily finite dimension. As it stands, E is too general to admit of interesting study. Thus, without any topology on E, questions like ‘Is vector space addition a continuous operation?’ or ‘Are linear maps continuous?’ are meaningless. We therefore wish to start by studying vector spaces with some additional structure such as a topology or metric. One such structure that the reader will have already encountered in the study of the vector spaces R, R2, R3 and C is the important notation of the length of a vector. The generalization of length to an arbitrary vector space is given by the following definition.
Definition 1.1.1
Let E be a vector space over the field K. A function
is called a norm on E if
1.
x
= 0 if and only if x = 0
2.
for all x, y ∈ E (‘triangle inequality’)
3.
for all x ∈ E and k ∈ K.
The pair (E,
) is then called a normed vector space.
A norm, then, is our generalization of length. The following proposition shows that the norm of a vector is, like length, always positive.
Proposition 1.1.2
Let (E,
) be a normed vector space. Then
x
0 for all x ∈ E...
Table of contents
Cover Page
Title Page
Copyright Page
Contents
Preface
CHAPTER 0: PRELIMINARIES
CHAPTER 1: LINEAR ALGEBRA AND NORMED VECTOR SPACES
CHAPTER 2: DIFFERENTIATION AND CALCULUS ON VECTOR SPACES
CHAPTER 3: THE INVERSE- AND IMPLICIT-FUNCTION THEOREMS
