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This is part two of a two-volume book on real analysis and is intended for senior undergraduate students of mathematics who have already been exposed to calculus. The emphasis is on rigour and foundations of analysis. Beginning with the construction of the number systems and set theory, the book discusses the basics of analysis (limits, series, continuity, differentiation, Riemann integration), through to power series, several variable calculus and Fourier analysis, and then finally the Lebesgue integral. These are almost entirely set in the concrete setting of the real line and Euclidean spaces, although there is some material on abstract metric and topological spaces. The book also has appendices on mathematical logic and the decimal system. The entire text (omitting some less central topics) can be taught in two quarters of 25–30 lectures each. The course material is deeply intertwined with the exercises, as it is intended that the student actively learn the material (and practice thinking and writing rigorously) by proving several of the key results in the theory.

Inhaltsverzeichnis

Frontmatter

Chapter 1. Metric spaces

Abstract
In Definition 6.1.5 we defined what it meant for a sequence \( \left( {x_{n} } \right)_{n = m}^{\infty } \) of real numbers to converge to another real number x; indeed, this meant that for every ε > 0, there exists an N ≥ m such that|x − x n | ≤ ε for all n ≥ N. When this is the case, we write lim n→∞  x n  = x.
Terence Tao

Chapter 2. Continuous functions on metric spaces

Abstract
In the previous chapter we studied a single metric space (X, d), and the various types of sets one could find in that space. While this is already quite a rich subject, the theory of metric spaces becomes even richer, and of more importance to analysis, when one considers not just a single metric space, but rather pairs (X, d X ) and (Y, d Y ) of metric spaces, as well as continuous functions f : X → Y between such spaces.
Terence Tao

Chapter 3. Uniform convergence

Abstract
In the previous two chapters we have seen what it means for a sequence \( \left( {x^{(n)} } \right)_{n = 1}^{\infty } \) of points in a metric space \( \left( {X,d_{X} } \right) \) to converge to a limit x; it means that \( \lim_{n \to \infty } d_{X} \left( {x^{(n)} ,x} \right) < \varepsilon \) or equivalently that for every \( \varepsilon > 0 \) there exists an N > 0 such that \( d_{X} \left( {x^{(n)} ,x} \right) < \varepsilon \) for all n > N. (We have also generalized the notion of convergence to topological spaces \( \left( {X, \fancyscript {F}} \right) \) but in this chapter we will focus on metric spaces.)
Terence Tao

Chapter 4. Power series

Abstract
We now discuss an important subclass of series of functions, that of power series. As in earlier chapters, we begin by introducing the notion of a formal power series, and then focus in later sections on when the series converges to a meaningful function, and what one can say about the function obtained in this manner.
Terence Tao

Chapter 5. Fourier series

Abstract
In the previous two chapters, we discussed the issue of how certain functions (for instance, compactly supported continuous functions) could be approximated by polynomials. Later, we showed how a different class of functions (real analytic functions) could be written exactly (not approximately) as an infinite polynomial, or more precisely a power series.
Terence Tao

Chapter 6. Several Variable Differential Calculus

Abstract
We shall now switch to a different topic, namely that of differentiation in several variable calculus. More precisely, we shall be dealing with maps f : R n R m from one Euclidean space to another, and trying to understand what the derivative of such a map is.
Terence Tao

Chapter 7. Lebesgue measure

Abstract
In the previous chapter we discussed differentiation in several variable calculus. It is now only natural to consider the question of integration in several variable calculus.
Terence Tao

Chapter 8. Lebesgue integration

Abstract
In Chapter 11, we approached the Riemann integral by first integrating a particularly simple class of functions, namely the piecewise constant functions. Among other things, piecewise constant functions only attain a finite number of values (as opposed to most functions in real life, which can take an infinite number of values). Once one learns how to integrate piecewise constant functions, one can then integrate other Riemann integrable functions by a similar procedure.
Terence Tao

Backmatter

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