Book contents
- Frontmatter
- Contents
- Preface
- 1 Introduction
- 2 The Real and Complex Numbers
- 3 Real and Complex Sequences
- 4 Series
- 5 Power Series
- 6 Metric Spaces
- 7 Continuous Functions
- 8 Calculus
- 9 Some Special Functions
- 10 Lebesgue Measure on the Line
- 11 Lebesgue Integration on the Line
- 12 Function Spaces
- 13 Fourier Series
- 14 * Applications of Fourier Series
- 15 Ordinary Differential Equations
- Appendix: The Banach-Tarski Paradox
- Hints for Some Exercises
- Notation Index
- General Index
2 - The Real and Complex Numbers
Published online by Cambridge University Press: 06 July 2010
- Frontmatter
- Contents
- Preface
- 1 Introduction
- 2 The Real and Complex Numbers
- 3 Real and Complex Sequences
- 4 Series
- 5 Power Series
- 6 Metric Spaces
- 7 Continuous Functions
- 8 Calculus
- 9 Some Special Functions
- 10 Lebesgue Measure on the Line
- 11 Lebesgue Integration on the Line
- 12 Function Spaces
- 13 Fourier Series
- 14 * Applications of Fourier Series
- 15 Ordinary Differential Equations
- Appendix: The Banach-Tarski Paradox
- Hints for Some Exercises
- Notation Index
- General Index
Summary
The previous chapter was somewhat informal. Starting in this chapter we develop the subject systematically and (usually) in logical order. This does not mean that every step in every chain of reasoning will be written out and referred back to the axioms or to results that have already been established. Such a procedure, though possible, is extremely tedious. The goal, rather, is to include enough results – and enough examples of reasoning – so that it may be clear how the gaps might be filled.
The Real Numbers
Our starting point is the real number system ℝ. This is a set that has two algebraic operations, addition and multiplication, and an order relation <. Let a, b denote arbitrary elements of ℝ. Addition associates to any pair a, b a real number denoted a + b, while multiplication associates a real number denoted a · b or simply ab. That < is a relation simply means that certain ordered pairs (a, b) of elements of ℝ are selected, and for these pairs (only) we write a < b. These operations and the order relation satisfy the following axioms, or conditions, in which a, b, c denote arbitrary elements of ℝ.
A1 (a + b) + c = a + (b + c).
A2a + b = b + a.
A3There is an element 0 such that, for all a, a + 0 = a.
A4For each a ∈ ℝ there is an element –a ∈ ℝ such that a + (–a) = 0.
M1 (ab)c = a(bc).
M2ab = ba.
M3There is an element 1 ≠ 0 in ℝ such that, for all a, a · 1 = a.
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- Information
- AnalysisAn Introduction, pp. 15 - 29Publisher: Cambridge University PressPrint publication year: 2004