Analysis I

Inhaltsverzeichnis

1. Frontmatter

2. Chapter 1. Introduction

   Terence Tao

   Abstract

   This text is an honors-level undergraduate introduction to real analysis: the analysis of the real numbers, sequences and series of real numbers, and real-valued functions. This is related to, but is distinct from, complex analysis, which concerns the analysis of the complex numbers and complex functions, harmonic analysis, which concerns the analysis of harmonics (waves) such as sine waves, and how they synthesize other functions via the Fourier transform, functional analysis, which focuses much more heavily on functions (and how they form things like vector spaces), and so forth. Analysis is the rigorous study of such objects, with a focus on trying to pin down precisely and accurately the qualitative and quantitative behavior of these objects. Real analysis is the theoretical foundation which underlies calculus, which is the collection of computational algorithms which one uses to manipulate functions.

3. Chapter 2. Starting at the Beginning: The Natural Numbers

   Terence Tao

   Abstract

   In this text, we will review the material you have learnt in high school and in elementary calculus classes, but as rigorously as possible. To do so we will have to begin at the very basics—indeed, we will go back to the concept of numbers and what their properties are. Of course, you have dealt with numbers for over ten years and you know how to manipulate the rules of algebra to simplify any expression involving numbers, but we will now turn to a more fundamental issue, which is: why do the rules of algebra work at all? For instance, why is it true that \(a(b+c)\) is equal to \(ab + ac\) for any three numbers a, b, c? This is not an arbitrary choice of rule; it can be proven from more primitive, and more fundamental, properties of the number system. This will teach you a new skill—how to prove complicated properties from simpler ones. You will find that even though a statement may be “obvious”, it may not be easy to prove; the material here will give you plenty of practice in doing so, and in the process will lead you to think about why an obvious statement really is obvious. One skill in particular that you will pick up here is the use of mathematical induction, which is a basic tool in proving things in many areas of mathematics.

4. Chapter 3. Set Theory

   Terence Tao

   Abstract

   Modern analysis, like most other subfields of modern mathematics, is concerned with numbers, sets, and geometry. We have already introduced one type of number system, the natural numbers. We will introduce the other number systems shortly, but for now we pause to introduce the concepts and notation of set theory, as they will be used increasingly heavily in later chapters. (We will not pursue a rigorous description of Euclidean geometry in this text, preferring instead to describe that geometry in terms of the real number system by means of the Cartesian co-ordinate system.)

5. Chapter 4. Integers and Rationals

   Terence Tao

   Abstract

   In Chapter 2 we built up most of the basic properties of the natural number system, but we have reached the limits of what one can do with just addition and multiplication. We would now like to introduce a new operation, that of subtraction, but to do that properly we will have to pass from the natural number system to a larger number system, that of the integers.

6. Chapter 5. The Real Numbers

   Terence Tao

   Abstract

   To review our progress to date, we have rigorously constructed three fundamental number systems: the natural number system \({{\textbf{N}}}\), the integers \({{\textbf{Z}}}\), and the rationals \({{\textbf{Q}}}\). We defined the natural numbers using the five Peano axioms and postulated that such a number system existed; this is plausible, since the natural numbers correspond to the very intuitive and fundamental notion of sequential counting. Using that number system one could then recursively define addition and multiplication and verify that they obeyed the usual laws of algebra. We then constructed the integers by taking formal differences of the natural numbers, \(a {\,\textemdash \,}b\). We then constructed the rationals by taking formal quotients of the integers, a//b, although we need to exclude division by zero in order to keep the laws of algebra reasonable. (You are of course free to design your own number system, possibly including one where division by zero is permitted; but you will have to give up one or more of the field axioms from Proposition 4.2.4, among other things, and you will probably get a less useful number system in which to do any real-world problems.).

7. Chapter 6. Limits of Sequences

   Terence Tao

   Abstract

   In the previous chapter, we defined the real numbers as formal limits of rational (Cauchy) sequences, and we then defined various operations on the real numbers. However, unlike our work in constructing the integers (where we eventually replaced formal differences with actual differences) and rationals (where we eventually replaced formal quotients with actual quotients), we did not completely finish the job of constructing the real numbers, because we never got around to replacing formal limits \({{\,\mathrm{{\text {LIM}}}\,}}_{n \rightarrow \infty } a_n\) with actual limits \(\lim _{n \rightarrow \infty } a_n\). In fact, we haven’t defined limits at all yet. This will now be rectified.

8. Chapter 7. Series

   Terence Tao

   Abstract

   Now that we have developed a reasonable theory of limits of sequences, we will use that theory to develop a theory of infinite series

   $$\begin{aligned} \sum _{n=m}^\infty a_n = a_m + a_{m+1} + a_{m+2} + \ldots .\end{aligned}$$

   But before we develop infinite series, we must first develop the theory of finite series.

9. Chapter 8. Infinite Sets

   Terence Tao

   Abstract

   We now return to the study of set theory, and specifically to the study of cardinality of sets which are infinite (i.e., sets which do not have cardinality n for any natural number n), a topic which was initiated in Section 3.6.

10. Chapter 9. Continuous Functions on 

    Terence Tao

    Abstract

    In previous chapters we have been focusing primarily on sequences. A sequence \((a_n)_{n=0}^\infty \) can be viewed as a function from \({{\textbf{N}}}\) to \({{\textbf{R}}}\), i.e., an object which assigns a real number \(a_n\) to each natural number n. We then did various things with these functions from \({{\textbf{N}}}\) to \({{\textbf{R}}}\), such as take their limit at infinity (if the function was convergent), or form suprema, infima, etc., or computed the sum of all the elements in the sequence (again, assuming the series was convergent).

11. Chapter 10. Differentiation of Functions

    Terence Tao

    Abstract

    We can now begin the rigorous treatment of calculus in earnest, starting with the notion of a derivative. We can now define derivatives analytically, using limits, in contrast to the geometric definition of derivatives, which uses tangents.

12. Chapter 11. The Riemann Integral

    Terence Tao

    Abstract

    In the previous chapter we reviewed differentiation—one of the two pillars of single variable calculus. The other pillar is, of course, integration, which is the focus of the current chapter. More precisely, we will turn to the definite integral, the integral of a function on a fixed interval, as opposed to the indefinite integral, otherwise known as the antiderivative. These two are of course linked by the Fundamental theorem of calculus, of which more will be said later.

13. Backmatter