Analysis II

Inhaltsverzeichnis

1. Frontmatter

2. Chapter 1. Metric Spaces

   Terence Tao

   Abstract

   In Definition 6.1.5 we defined what it meant for a sequence \((x_n)_{n=m}^\infty \) of real numbers to converge to another real number x; indeed, this meant that for every \({\varepsilon }> 0\), there exists an \(N \ge m\) such that \(|x-x_n| \le {\varepsilon }\) for all \(n \ge N\). When this is the case, we write \(\lim _{n \rightarrow \infty } x_n = x\).

3. Chapter 2. Continuous Functions on Metric Spaces

   Terence Tao

   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 \(f:X \rightarrow Y\) between such spaces.

4. Chapter 3. Uniform Convergence

   Terence Tao

   Abstract

   In the previous two chapters we have seen what it means for a sequence \((x^{(n)})_{n=1}^\infty \) of points in a metric space \((X,d_X)\) to converge to a limit x; it means that \(\lim _{n \rightarrow \infty } d_X(x^{(n)}, x) = 0\), or equivalently that for every \({\varepsilon }> 0\) there exists an \(N > 0\) such that \(d_X(x^{(n)},x) < {\varepsilon }\) for all \(n > N\).

5. Chapter 4. Power Series

   Terence Tao

   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.

6. Chapter 5. Fourier Series

   Terence Tao

   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.

7. Chapter 6. Several Variable Differential Calculus

   Terence Tao

   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 \(f:{{\textbf{R}}}^n \rightarrow {{\textbf{R}}}^m\) from one Euclidean space to another, and trying to understand what the derivative of such a map is.

8. Chapter 7. Lebesgue Measure

   Terence Tao

   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. The general question we wish to answer is this: given some subset \(\Omega \) of \({{\textbf{R}}}^n\), and some real-valued function \(f:\Omega \rightarrow {{\textbf{R}}}\), is it possible to integrate f on \(\Omega \) to obtain some number \(\int _\Omega f\)? (It is possible to consider other types of functions, such as complex-valued or vector-valued functions, but this turns out not to be too difficult once one knows how to integrate real-valued functions, since one can integrate a complex or vector-valued function, by integrating each real-valued component of that function separately.)

9. Chapter 8. Lebesgue Integration

   Terence Tao

   Abstract

   In Chap. 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.

10. Backmatter