A Course in Analysis

The following sections are included:

• Preface
• Introduction
• Contents
• List of Symbols

At this stage of our Course much has already been learnt about integration. In Part 1 we introduced the Riemann integral for real-valued functions of one real variable, in Part 2 we provided the underlying theory including all proofs. Then, in Part 4, we investigated integrals of real-valued functions of several real variables, followed by line and surface integrals of vector fields in Part 5. A motivation for introducing integrals was the problem to determine the area (volume) of a figure bounded by the graphs of certain functions. A further central topic was the fundamental theorem of calculus which links the derivative of a function to the integral. Powerful as these classical results are, eventually they lead to many open questions; here are some of these:

• We can integrate certain discontinuous functions, but we cannot (yet) prove the fundamental theorem for them. It would be desirable to have necessary and sufficient conditions the fundamental theorem to hold.
• We can define the length of a rectifiable curve and integrals over rectifiable curves, but we lack an analogous theory for surfaces or more generally hypersurfaces.
• How do we define a volume integral for functions f : G → ℝ, G ⊂ ℝⁿ, without depending on topological notions, i.e. ∂G?
• Can we derive more satisfactory results for interchanging limits (continuity, differentiability, sequences) and integrals?

While in the last chapter we introduced the framework for eventually resolving our problem of defining a measure and integral for real-valued functions defined on suitable domains in ℝⁿ, we still cannot apply this framework to this concrete situation, i.e.(ℝⁿ, ℬ⁽ⁿ⁾). The measure λ⁽ⁿ⁾ we are seeking on ℬ⁽ⁿ⁾ should satisfy

The Lebesgue-Borel measure and its extension to the Lebesgue measure is a fundamental object of analysis and in this chapter we investigate more of its properties and extensions. Most of all we are interested in “geometric” properties, for example invariance properties. We need

Lemma 3.1. Let B ∊ ℬ⁽ⁿ⁾ be a Borel set, x₀ ∊ ℝⁿ and T ∊ O(n). Then x + B and T(B) are Borel sets too…

We have already introduced measurable mappings f : Ω → Ω′ where (Ω, 𝒜) and (Ω′, 𝒜′) are two measurable spaces. These are mappings having the property that the pre-image of 𝒜′-measurable sets are 𝒜-measurable, i.e. f^-1(A′)∊ 𝒜 for A′ ∊ 𝒜′, see Definition 1.15. In this chapter we want to investigate measurable mappings further, but first let us recollect what is already known to us:

• the composition of measurable mappings is measurable (Remark 1.16.C);
• f is measurable if and only if f⁻¹(E′) ∊ 𝒜 for a generator E′ of 𝒜′ (Lemma 1.17);
• for measurable spaces (Ω_j, 𝒜_j), j ∊ I, and mappings f_j : Ω→Ω_j exists a smallest σ−field in Ω such that all f_j become measurable (Example 1.14 and Remark 1.16.A);
• if f :Ω→Ω′ is measurable and µ is a measure on (Ω,𝒜) then the image f(µ) is a well defined measure on (Ω′,𝒜′) (Theorem 3.7);
• forming image measures is transitive (Corollary 3.9).

Let (Ω, 𝒜, μ) be a measure space and denote by S(Ω) the set of all simple functions u : Ω→ℝ. Suppose that {γ1, …, γK} ⊂[0, ∞) is the range of u and γ_j ≠ γ_k for j ≠ k. In this case the sets C_l := {u = γ}, l = 1, … , K, are mutually disjoint and we can define the (μ–)integral of u by

In the last chapter we discussed integration with respect to an image measure, compare for example with the statement of Corollary 5.32, and we looked at measures being invariant under transformations and the behaviour of corresponding integrals. One purpose of this chapter is to study the behaviour of integrals over Borel sets G⊂ℝⁿ with respect to the measure λ⁽ⁿ⁾ under suitable transformations. We start with Example 6.1. For T∊ GL(n;ℝ) we know from Theorem 3.11 that

Let (Ω 𝒜, μ) be a measure space and (u_k)_k∊ℕ, u_k : Ω → ℝ, be a sequence of measurable functions. We want to discuss several different types of convergence of the sequence (u_k)_k∊ℕ to a function u : Ω→ ℝ. As we will see later in this chapter we can often replace ℝ by ℂ, ℝⁿ or even a normed vector space (V, k.k), and sometimes we may replace ℝ by $\overline{ℝ}$. Independent of 𝒜 and μ we have of course the notion of pointwise convergence: a sequence of functions u_k : Ω→ ℝ converges pointwisely to u : Ω→ ℝ if for every ω∊Ω and every ∊ > 0 there exists N = N(∊, ω)∊ℕ such that k≥ N(∊, ω) implies ∣u_k(ω) –u(ω)∣ < ∊. In this definition we can obviously replace (ℝ, ∣.∣) by a normed ℝ-vector space (V, ‖.‖), i.e. u_k : Ω→V converges pointwisely to u : Ω → V if for every ω∊Ω and every ∊> 0 there exists N = N(∊, ω) such that k ≥ N(∊, ω) implies ‖u_k(ω) − u(ω)‖ < ∊…

In this chapter we collect various applications of the convergence results proved in the last chapter. Some are just useful tools, others are important statements in their own right. We will investigate the relations between the Lebesgue and the Riemann integral for domains in ℝⁿ. Finally we also discuss how we can pass from the spaces ℒ^p(Ω) to the Banach spaces L^p(Ω) and therefore prepare some topics needed in functional analysis…

Before introducing the Riemann integral for certain domains in ℝⁿ we studied in Chapter II.17 iterated integrals and then, in Chapter II.18 their relation to integrals over a hyper-rectangle in ℝⁿ, see Theorem II.18.19. We can interpret a hyper-rectangle as a product of intervals and an interesting question is whether we can find a natural measure on the product of two or finitely many measure spaces such that the integral with respect to this “product measure” will be for certain functions an iterated integral. In Chapter 1 we discussed product δ-fields and their generators and we will use the notations introduced there, see especially Definition 1.19 and Theorem 1.20…

In Problem 10 of Chapter II.22 we briefly discussed the convolution for continuous functions with compact support, i.e. the function

Let f : [a, b] → ℝ be a Borel measurable function. We want to investigate questions such as: for which points x ∊ [a, b] is f differentiable, or if f is integrable, when is $x\mapsto {\displaystyle \int {}_{\left[a,x\right]}}f\left(t\right)\lambda {\left(1\right)}^{}\left(\text{d}t\right)$ differentiable, and does the fundamental theorem hold?…

In this chapter we want to discuss three topics with some of their applications: the theorem of Sard; the theorem of Lusin; and Kolmogorov’s version of the Arzela-Ascoli theorem for L^p-spaces. In some discussions we will need auxilliary results (mainly from point set topology) which we will quote but not prove. The main idea of this chapter is to present some useful results and to indicate that the theory developed so far has far reaching consequences in quite different, and maybe unexpected fields of mathematics…

Although we assume that the reader is familiar with the complex numbers 𝕔 from studies about algebra or geometry, we want to discuss in some detail the algebraic properties of 𝕔 as well as the fact that 𝕔 equipped with the modulus or absolute value is metric in a complete metric space. In fact 𝕔 is also a complete algebraic field in the sense that every polynomial factorises into linear factors, but for this we will give a proof much later in Chapter 23 and our proof will be analytic not algebraic. This chapter also serves to fix notations and to make a start with getting used to the traditional façon de parler in the theory of complex functions of a complex variable…

Part 7 is essentially devoted to functions defined on some subset D ⊂ 𝕔 with values in 𝕔. The fact that 𝕔 is a field allows us to consider difference quotients $\frac{f\left(z\right)-f\left(z0\right)}{z-z0}$ and hence complex differentiation which is significantly different from differentiating functions g : G → ℝ², G ⊂ ℝ²…

The preparations of Chapter 13 allow us to start immediately with the investigation of complex-valued functions defined on a subset G ⊂ 𝕔 and nowadays many textbooks on this topic follow the same approach. However complex analysis is also a rather geometric theory and we prefer to collect early in a more or less coherent manner interpretations and results linking geometry and algebra in 𝕔…

In Chapter 14 we discussed complex-valued functions defined on an arbitrary non-empty set. When specifying this set to a subset D ⊂ 𝕔 we can of course apply the earlier results. We denote points in D by z = x+iy and it follows that every f : D → 𝕔 admits a decomposition

Let f : D → 𝕔 be a function defined on a non-empty open subset of 𝕔. For z, z₀ ∊ D, z ≠ z₀, the difference quotient

In this chapter we want to discuss some of the so called elementary transcendental functions defined on complex domains. We start with the exponential function exp defined by the power series

Line integrals of holomorphic and related functions turn out to be very helpful tools in investigating these functions. In fact they lead to completely new insights, including, surprisingly, about the relation between analysis and topology. For understanding these relations we first need to study in more detail the notion of a connected set in the plane. However, since many ideas and results hold for more general topological spaces and since we want to make use of these results when we study differential geometry or differentiable manifolds later on, we discuss them here either in the context of topological spaces or at least in the context of metric spaces. But there will also be a few results especially related to the plane. Note that from the topological or metrical point of view ℂ and ℝ² are identical, but sometimes we may prefer to use complex numbers or complex-valued functions for the formulation of topological statements in the plane…

For a parametrized curve γ : [a, b] → ℂ with tr γ⊂ D, D ⊂ ℂ a domain, we want to define for complex-valued functions f : D→ ℂ the line integral ${\displaystyle \underset{\gamma }{\int }f\left(z\right)dz}$. While in the case of holomorphic functions the properties of ${\displaystyle \underset{\gamma }{\int }f\left(z\right)dz}$ will turn out to be quite different to those of general line integrals (in ℝⁿ), the basic definition of the line integral ${\displaystyle \underset{\gamma }{\int }f\left(z\right)dz}$ is similar, in fact it is an adaptation of the theory developed in Chapter II.15 where we now identify ℝ² with ℂ. Therefore we start by recollecting definitions and results from Chapter II.15 but now we will use complex notation. From this chapter on we will also use the standard notation from complex variable theory and we will often write γ for the curve (as mapping) as well as for its trace…

The results in this Chapter can be viewed as the starting point as well as the essence of Cauchy’s approach to complex variable theory. We start with a more or less heuristic (but still correct) consideration. Let G ⊂ ℂ ≅ ℝ² be a domain for which Green’s theorem in the form of (II.26.13) holds. In particular ∂ G is the trace of a parametric curve and G can be considered as a normal domain with respect to both coordinate axes. Since we identify ℝ² with ℂ the canonical basis in ℝ² can be identified with {1, i}, a vector z has components z = x + iy and f : G → ℂ is interpreted as vector field f(z) = u (x, y) + i v (x, y). Suppose that f is in a neighbourhood of G holomorphic, i.e. f′(z) exists. We assume in addition that f′ is continuous implying that u_x, u_y, v_x and vy are continuous too. Now Green’s theorem, i.e. (II.26.13) yields…

Let us summarize what we know about holomorphic functions so far…

In this chapter we want to discuss several properties of holomorphic functions, including some of their global mapping properties, as well as properties of their range, the question of how we can approximate holomorphic functions, and infinite products of holomorphic functions. We start with some mapping properties and we need the result below as preparation…

Rational functions $R=\frac{P}{Q},P$ and Q are polynomials, are defined on $ℂ\backslash \{z\in ℂ|Q\left(z\right)=0\}$and in this set they are holomorphic. Since Q has only a finite number of zeroes, in the case where Q is of degree m the set $\{z\in ℂ|Q\left(z\right)=0\}$has by the fundamental theorem of algebra at most m elements, the function R is holomorphic in ℂ except at a finite number of points. For reasons which will become clear later we write

So far when working with closed curves we emphasised that they were simply closed, in particular for the circle |z₀ - z|= r we used the standard (anticlockwise) parametrization $t\mapsto z\left(t\right)=z_0+r{e}^{it},t\in [0,2\pi ].$ A very useful observation was that with this parametrization we have

This chapter is devoted to some special functions which are of central importance in many branches of mathematics. In Volume I we encountered the Γ-function and related functions such as the beta-function, but on that occasion we considered these functions as real-valued depending on real variables. Now we want to study them as complex-valued functions of a complex variable…

The purpose of this chapter is to indicate the existence of an entirely new area of Mathematics, i.e. the theory of elliptic integrals and elliptic functions. We cannot even start to explore this in our Course, however the reader should learn about its existence, its historical importance and that today many branches of Mathematics originate from it. In our treatment of the history of analysis we will address some of the major developments of the theory of elliptic functions and the influence they had and still have…

Our approach to complex-valued functions of a complex variable is in some sense an analytic approach putting properties of holomorphic functions on the centre stage. However on several occasions we encountered the need to think more (complex-) geometrical. In fact, once basic notions and tools have been introduced, we can look more at topological and most of all geometric or metric mapping properties of these functions. Here the aspect of mapping sets onto sets becomes more prominent. One such result is Theorem 23.2 stating that holomorphic functions are open mappings, i.e. they map open sets onto open sets, provided they are not constant. In this chapter we want to look at simply connected open sets in the plane and study their biholomorphic images…

In this chapter we discuss some properties of power series of several variables, complex or real. We do not intend to give an introduction to the theory of complex-valued functions of several variables, this is beyond our Course. However already when looking at real-valued functions of several real variables, we stopped short of treating Taylor series which should be of course power series. We have seen the very close relationship between power series of one real variable and of one complex variable. Now we will see some similarity in several variables, but convergence domains (and results) will be quite different. This is best understood in several complex variables although in [51] a readable treatment of the real case is available, and it also indicates why the theory of differentiable functions from some set G ⊂ ℂⁿ to ℂ should be expected to have quite different features than the one-dimensional theory. In our presentation we make use of [29] and [66]…

The following sections are included:

• Appendix I: More on Point Set Topology
• Appendix II: Measure Theory, Topology and Set Theory
• Appendix III: More on Möbius Transformations
• Appendix IV: Bernoulli Numbers

The following sections are included:

• Solutions to Problems of Part 6
• Solutions to Problems of Part 7

The following sections are included:

• References
• Mathematicians Contributing to Analysis (Continued)
• Subject Index