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In this chapter, we generalize to more abstract settings two key properties of ]> : the Euclidean norm of a vector and the dot product of two vectors. The generalizations, norms and inner products, respectively, are set in a general vector space. Many of our applications will be set in this framework. The basic notions of topology go through with almost no change in the definitions. However, some theorems can be quite different. For example, being closed and bounded is not sufficient to imply compactness in an infinite-dimensional normed vector space, such as C[a,b]. [ABSTRACT FROM AUTHOR]

A natural problem is to take a wave output and decompose it into its harmonic parts. Engineers are able to do this with an oscilloscope. A real difficultly occurs when we try to put the parts back together. Mathematically, this amounts to summing up the series obtained from decomposing the original wave. In this chapter, we examine this delicate question: Under what conditions does a Fourier series converge? We start by looking at the behaviour of the Fourier coefficients. It turns out that they go to zero; and the smoother the function, the faster they go. Then we turn to the more subtle questions of pointwise and uniform convergence. The idea of kernel functions, analogous to the Poisson kernel from the previous chapter, provides an elegant method for understanding these notions of convergence. Then we turn to the L2 norm, where there is a very clean answer. Nice applications of this include the isoperimetric inequality and sums of various interesting series. Finally, we consider applications to polynomial approximation. [ABSTRACT FROM AUTHOR]

Optimization is a central theme of applied mathematics that involves minimizing or maximizing various quantities. This is an important application of the derivative tests in calculus. In addition to the first and second derivative tests of one-variable calculus, there is the powerful technique of Lagrange multipliers in several variables. This chapter is concerned with analogues of these tests that are applicable to functions that are not differentiable. Of course, some different hypothesis must replace differentiability, and this is the notion of convexity. It turns out that many applications in economics, business, and related areas involve convex functions. As in other chapters of this book, we concentrate on the theoretical underpinnings of the subject. The important aspect of constructing algorithms to carry out our program is not addressed. However, the reader will be well placed to read that material. Results from both linear algebra and calculus appear regularly. The study of convex sets and convex functions is a comparatively recent development. Although convexity appears implicitly much earlier (going back to work of Archimedes, in fact), the first papers on convex sets appeared at the end of the nineteenth century. The main theorems of this chapter, characterizations of solutions of optimization problems, first appeared around the middle of the twentieth century. Starting in the 1970s, there has been considerable work on extending these methods to nonconvex functions. [ABSTRACT FROM AUTHOR]

The space ]> is the right setting for many problems in real analysis. For example, in many situations, functions of interest depend on several variables. This puts us into the realm of multivariable calculus, which is naturally set in ]> . We will study normed vector spaces further in Chapter 7, building on the properties and concepts we study here. The space ]> is the most important normed vector space, after the real numbers themselves. [ABSTRACT FROM AUTHOR]

In this chapter we develop an important variation on Fourier series, replacing the sine and cosine functions with new families of functions, called wavelets. The strategy is to construct wavelets so that they have some of the good properties of trig functions but avoid the failings of Fourier series that we have seen in previous chapters. With such functions, we can develop new versions of Fourier series methods that will work well for problems where traditional Fourier series work poorly. What are the good properties of trig functions? First and foremost, we have an orthogonal basis in L2, namely the set of functions sin(nx) and cos(nx) as n runs over ]> . This leads to the idea of breaking up a wave into its harmonic constituents, as the sine and cosine functions appear in the solution of the wave equation. We want to retain some version of this orthogonality. [ABSTRACT FROM AUTHOR]

Fourier series were first developed to solve partial differential equations that arise in physical problems, such as heat flow and vibration. We will look at the physics problem of heat flow to see how Fourier series arise and why they are useful. Then we will proceed with the solution, which leads to a lot of very interesting mathematics. We will also see that the problem of a vibrating string leads to a different PDE that requires similar techniques to solve. While these problems sound very applied, the infinite series that arise as solutions forced mathematicians to delve deeply into the foundations of analysis. When d΄Alembert proposed his solution for the motion of a vibrating string in 1754, there were no clear, precise definitions of limit, function, or even of the real numbers–all things taken for granted in most calculus courses today. D΄Alembert΄s solution has a closed form, and thus did not really challenge deep principles. However, the solution to the heat problem that Fourier proposed in 1807 required notions of convergence that mathematicians of that time did not have. Fourier won a major prize in 1812 for this work, but the judges, Laplace, Lagrange, and Legendre, criticized Fourier for lack of rigour. Work in the nineteenth century by many now famous mathematicians eventually resolved these questions by developing the modern definitions of limit, continuity, and uniform convergence. These tools were developed not because of some fetish for finding complicated things, but because they were essential to understanding Fourier series. [ABSTRACT FROM AUTHOR]

In this chapter, we apply analysis to the study of ordinary differential equations, generally called DEs or ODEs. Ordinary is used to indicate differential equations of a single variable, in contrast to partial differential equations (PDEs), in which several variables, and hence partial derivatives, appear. We will see some PDEs in the chapters on Fourier series, Chapters 13 and 14. Most introductory courses on differential equations present methods for solving DEs of various special types. We will not be concerned with those techniques here except to give a few pertinent examples. Rather we are concerned with why differential equations have solutions, and why these solutions are or are not unique. This topic, crucial to a full understanding of differential equations, is often omitted from introductary courses because it requires the tools of real analysis. In particular, we will require the Banach Contraction Principle (11.1.6) which was established in the previous chapter. [ABSTRACT FROM AUTHOR]

Suppose we wish to describe some physical system. The dynamical systems approach considers the space X of all possible states of the system–think of a point x in X as representing physical data. We will assume that X is a subset of some normed vector space, often ]> . The evolution of the system over time determines a function T of X into itself that takes each state to a new state, one unit of time later. [ABSTRACT FROM AUTHOR]

This chapter introduces some of the essentials of approximation theory, in particular approximating functions by ˵nice″ ones such as polynomials. In general, the intention of approximation theory is to replace some complicated function with a new function, one that is easier to work with, at the price of some (hopefully small) difference between the two functions. The new function is called an approximation. There are two crucial issues in using an approximation: first, how much simpler is the approximation? and second, how close is the approximation to the original function? Deciding which approximation to use requires an analysis of the trade-off between these two issues. Of course, the answers to these two questions depend on the exact meanings of simpler and close, which vary according to the context. In this chapter, we study approximations by polynomials. Close is measured by some norm. We concentrate on the uniform norm, so that a polynomial approximation should be close to the function everywhere on a given interval. Approximations are closely tied to the notions of limit and convergence, since a sequence of functions approximating a function f to greater and greater accuracy converges to f in the norm used. Different approximation schemes correspond to different norms. [ABSTRACT FROM AUTHOR]

This text focuses on subsets of a normed space, since this is the natural setting for most of our applications. In this chapter, we introduce an apparently more general framework, metric spaces, and some new ideas that are somewhat more advanced. They play an occasional role in the advanced sections of the applications. [ABSTRACT FROM AUTHOR]

There are several reasonable definitions for the limit of a sequence of functions. Clearly the entries of the sequence should approximate the limit function f to greater and greater accuracy in some sense. But there are different ways of measuring the accuracy of an approximation, depending on the problem. Different approximation schemes generally correspond to different norms, although not all convergence criteria come from a norm. In this section, we consider two natural choices and see why the stronger notion is better for many purposes. [ABSTRACT FROM AUTHOR]

In this chapter, we examine the mathematical foundations of differentiation and integration. The theorems of this chapter are useful not only to make calculus work but also for studying functions in many other contexts.We do not spend any time on the important applications that typically appear in courses devoted to calculus, such as optimization problems. Rather we will highlight those aspects that either depend on or apply to results in real analysis. Since we assume that the reader has already seen calculus, we dive right in with key definitions, assuming that the motivating examples are familiar. [ABSTRACT FROM AUTHOR]

The main purpose of this chapter is to introduce the notion of a continuous function. Continuity is a basic notion of analysis. It is only with continuous functions that there can be any reasonable approximation or estimation of values at specific points. Most physical phenomena are continuous over most of their domain. The ideas we study in this chapter are sufficiently powerful that even discrete phenomena are sometimes best understood using continuous approximations where these ideas can be used. [ABSTRACT FROM AUTHOR]

We turn now to the problem of adding up an infinite series of numbers. As we shall quickly see, this is really no different from dealing with the sequence of partial sums of the series. However, there are tests for convergence that are more conveniently expressed for series than for sequences. [ABSTRACT FROM AUTHOR]

Doing analysis in a rigorous way starts with understanding the properties of the real numbers. Readers will be familiar, in some sense, with the real numbers from studying calculus. A completely rigorous development of the real numbers requires checking many details. We attempt to justify one definition of the real numbers without carrying out the proofs. Intuitively, we think of the real numbers as the points on a line stretching off to infinity in both directions. However, to make any sense of this, we must label all the points on this line and determine the relationship between them from different points of view. First, the real numbers form an algebraic object known as a field, meaning that one may add, subtract, and multiply real numbers and divide by nonzero real numbers. There is also an order on the real numbers compatible with these algebraic properties, and this leads to the notion of distance between two points. [ABSTRACT FROM AUTHOR]

Since we use results from calculus and linear algebra regularly, we review the key definitions and theorems here. If something seems unfamiliar, reviewing this material would be wise.We list a few good books in each subject in the bibliography. For the theoretical part of calculus, there is a detailed development in Chapter 6 and in the supplementary materials for this book available online. Finally, we give a brief treatment of equivalence relations. [ABSTRACT FROM AUTHOR]