Murray H. Protter

Functions on Metric Spaces; Approximation

We developed many of the basic properties of metric spaces in Chapter 6. A complete metric space was defined in Chapter 13 and we saw the importance of such spaces in the proof of the fundamental fixed point theorem (Theorem 13.2). This theorem, which has many applications, was used to prove the existence of solutions of ordinary differential equations and the Implicit function theorem. We now discuss in more detail functions whose domain is a metric space, and we prove convergence and approximation theorems which are useful throughout analysis.

Differentiation in ℝ N

There are two principal extensions to ℝ N of the theory of differentiation of real-valued functions on ℝ1. In this section, we develop the natural generalization of ordinary differentiation discussed in Chapter 4 to partial differentiation of functions from ℝ N to ℝ1. In Section 7.3, we extend the ordinary derivative to the total derivative.

Functions Defined by Integrals; Improper Integrals

The solutions of problems in differential equations, especially those which arise in physics and engineering, are frequently given in terms of integrals. Most often either the integrand of the integral representing the solution is unbounded or the domain of integration is an unbounded set. In this chapter we develop rules for deciding when it is possible to interchange the processes of differentiation and integration—commonly known as differentiation under the integral sign. When the integrand becomes infinite at one or more points or when the interval of integration is infinite, a study of the convergence of the integral is needed in order to determine whether or not the differentiation process is allowable. We establish the required theorems for bounded functions and domains in this section and treat the unbounded case in Sections 11.2 and 11.3.

Elementary Theory of Integration

The reader is undoubtedly familiar with the idea of integral and with methods of performing integrations. In this section we define integral precisely and prove the basic theorems which justify the processes of integration employed in a first course in calculus.

Contraction Mappings, Newton’s Method, and Differential Equations

The main result of this section is a simple theorem which proves to be useful in the solution of algebraic and differential equations. It will also be used to prove the important Implicit function theorem in Chapter 14.

Implicit Function Theorems and Lagrange Multipliers

Suppose we are given a relation in ℝ2 of the form

The Riemann—Stieltjes Integral and Functions of Bounded Variation

F\left( {x,y} \right) = 0.

Elementary Theory of Metric Spaces

(14.1) Then to each value of x there may correspond one or more values of y which satisfy (14.1)—or there may be no values of y which do so. If I = x: x0 − h < x < x0 + h is an interval such that for each x - I there is exactly one value of y satisfying (14.1), then we say that F(x, y) = 0 defines y as a function of x implicitly on I. Denoting this function by f, we have F[x, f(x)] = 0 for x on I.