main-content

Über dieses Buch

Science students have to spend much of their time learning how to do laboratory work, even if they intend to become theoretical, rather than experimental, scientists. It is important that they understand how experiments are performed and what the results mean. In science the validity of ideas is checked by experiments. If a new idea does not work in the laboratory, it must be discarded. If it does work, it is accepted, at least tentatively. In science, therefore, laboratory experiments are the touchstones for the acceptance or rejection of results. Mathematics is different. This is not to say that experiments are not part of the subject. Numerical calculations and the examina­ tion of special and simplified cases are important in leading mathematicians to make conjectures, but the acceptance of a conjecture as a theorem only comes when a proof has been constructed. In other words, proofs are to mathematics as laboratory experiments are to science. Mathematics students must, therefore, learn to know what constitute valid proofs and how to construct them. How is this done? Like everything else, by doing. Mathematics students must try to prove results and then have their work criticized by experienced mathematicians. They must critically examine proofs, both correct and incorrect ones, and develop an appreciation of good style. They must, of course, start with easy proofs and build to more complicated ones.

Inhaltsverzeichnis

Chapter 0. Some Ideas of Logic

Abstract
The dependence of mathematics on logic is obvious. We use reasoning processes in mathematics to prove results, and logic is concerned with reasoning. However, when someone studies mathematics, he does not first study logic in order to learn to think correctly. Rather, he jumps into mathematics, perhaps with high school geometry, and learns to prove things by actually doing proofs. Logic comes into his education only when there seems to be something doubtful or obscure that needs clarification. A person who is working with some parts of mathematics, like foundational studies, where common sense does not provide enough precision, needs the more finely-tuned results of logic. However, in most of the undergraduate mathematics courses you can get by with informal reasoning and common sense. There are a few exceptions. It is necessary to have a clear understanding of some of the vocabulary of logic as used in mathematics and of the logic underlying the idea of a mathematical proof. I will try to clarify some of these points in this chapter. You should read it over quickly and refer back to it when the need arises. At the end of the chapter I will give you an opportunity to use what you have learned by asking you to construct some simple proofs.
Robert B. Reisel

Chapter I. Sets and Mappings

Abstract
In this chapter you will learn some of the basic ideas of set theory, particularly those associated with mappings or functions. These will be used extensively in the following chapters and, indeed, in all of mathematics. This is not a systematic treatment of set theory. It omits many essential ideas, such as, infinity, cardinal and ordinal numbers, and foundational problems, that are not directly needed in this book. I assume that you have had some experience with the use of sets, so I do not give many examples of the familiar terms. Of course, you should not look in other books to find proofs of the theorems and the exercises of this chapter, because that would defeat the purpose of the book. However, if you want to see further discussion of the topics taken up here, you can look at books in set theory.
Robert B. Reisel

Chapter II. Metric Spaces

Abstract
In this chapter you will begin the study of metric spaces, the main topic of this book. A metric space is essentially a set in which it is possible to speak of the distance between any two of its elements. (Mathematicians usually refer to the set as a “space” and to its elements as “points” to emphasize the geometrical aspect of this study.) The theory of metric spaces is the general theory which underlies real analysis (calculus), complex analysis, multidimensional calculus and many other subjects.
Robert B. Reisel

Chapter III. Mappings of Metric Spaces

Abstract
In the last chapter you learned about some of the “geometry” of metric spaces. These geometrical ideas are more general than those you studied in high school, such as the congruence or similarity of triangles. They underlie all of geometry and are fundamental for the study of analysis, the branch of mathematics that develops and extends the ideas of calculus. In calculus you studied derivatives and integrals, which in turn were based on the idea of limits. It is possible to define limits in a metric space, but I will examine a more basic concept, that of continuous mappings of metric spaces.
Robert B. Reisel

Chapter IV. Sequences in Metric Spaces

Abstract
Sequences of real numbers and their limits play a large role in calculus and in applied mathematics. In this chapter you will study sequences in the broader context of metric spaces, but the results, of course, apply to sequences of real numbers. The chapter begins with a careful study of sequences in general and then passes to metric spaces where the concept of limit can be introduced. This is followed by a look at how sequences are related to metric properties, like closure or continuity. Finally, there is a brief study of Cauchy sequences and complete metric spaces.
Robert B. Reisel

Chapter V. Connectedness

Abstract
In this chapter you will study the concept of connectedness. Roughly speaking, the idea is to distinguish those spaces and sets that split up into several pieces from those that are all one piece. This can be done in several different ways and I will take up just one such criterion. After the general ideas have been worked out, you will look at the situation on the real line and prove an important theorem — the Intermediate Value Theorem.
Robert B. Reisel

Chapter VI. Compactness

Abstract
In a metric space there are usually sequences which do not have cluster points, but there are some metric spaces in which every sequence does have a cluster point. Such spaces are said to be sequentially compact. A related concept is that of compactness and a metric space with this property is called a compact space. As you will see, these two concepts are equivalent for metric spaces, but since they are not equivalent for the more general case of topological spaces, it is customary to study them separately. After you have learned about some of the properties of compact and sequentially compact metric spaces and have proved that they are equivalent, you will apply the results to the metric space R and derive some very important theorems of analysis.
Robert B. Reisel

Backmatter

Weitere Informationen