Advanced Calculus for Economics and Finance

This chapter summarises various preliminary results that the reader should be familiar with. It begins by introducing the notion of relation and function in set theory. These concepts help in the definition of equivalence and order relations, which, in turn, are used to define supremum, infimum, and upper and lower bounds. Then a review of countable sets and field structure follows. This provides a guide to the fundamental properties of real numbers, which are required as background to subsequent parts of this book. The chapter ends by introducing a few notions and results from linear algebra which will be useful in the discussion of normed spaces and multivariate functions. The treatment of the matter is intended to be a brief overview, and a small number of examples are provided. The reader is advised to initially pass over this chapter and return if any of the definitions or concepts in the following chapters is unclear.

This chapter describes topological spaces with a general, axiomatic, and abstract approach. The discussion is initially far from the space of real numbers which students are likely to be familiar with. The purpose of this treatment is to guide the reader through the use of a deductive and theorem-proving approach to analyse the properties of topological spaces. This avoids the naive geometric intuition which is often adopted in textbooks and is based on the more demanding notion of distance. As the analysis progresses, references to the fundamental theorems in \(\mathbb {R}\) are collected in specific sections. In this way, the reader can realise the connection of the abstract treatment with the results typically presented in a basic course in real calculus. Specifically, we will prove the Bolzano–Weierstrass and Heine–Borel theorems for real numbers and derive some very useful properties of continuous functions using only the topological structure. The observation that continuous functions map compact sets to compact sets easily translates into the Weierstrass extreme value theorem. Since continuity is among the most relevant topics in calculus, I will spend some time discussing the equivalence between the “global” notion of continuity defined using open sets and the “local” notion defined using the limit of the function. While this chapter is essentially self-contained, previous knowledge of basic notions of set theory, such as the properties of unions and intersections, is assumed.

The topological spaces introduced in Chapter 2 allowed us to discuss the notion of proximity and introduce basic concepts of modern analysis, such as compactness, connectedness, and continuity. We have studied extensively one specific example, the set of real numbers. In this chapter, we extend the discussion of these topics by introducing the concept of metric space. A metric space is essentially a set of elements, or points, together with a real-valued nonnegative function, which measures the distance between them. The metric structure brings our discussion closer to the typical spaces of mathematical calculus. With the notion of distance, the formalisation of the idea of proximity goes beyond the qualitative approach of topology, so to speak, to take a quantitative flavour. It turns out that the metric spaces are Hausdorff first-countable topological spaces. Thus, many results that we have derived in Chapter 2 can be fruitfully applied to their study. However, some key aspects of metric spaces are investigated using the notion of sequence, introduced in Chapter 5, so they will be postponed to Section 5.2.

This chapter introduces normed spaces, which are an essential tool in the study of real functions conducted in Chaps. 6 and 7. Normed spaces are a special case of metric spaces, and their study is an excellent exercise in applying a good share of the theory of topological and metric spaces developed in Chaps. 2 and 3. Normed spaces are linear spaces, and an important distinction is whether their dimension is finite or infinite. The case of finite-dimensional normed spaces is the one in which we are mainly interested in these notes. We will see that they share a large amount of properties with the set of real numbers \(\mathbb {R}\). In particular, we will extend the Bolzano–Weierstrass and Heine–Borel theorems to finite-dimensional normed spaces. When possible, the results will be derived in total generality, and a few examples of infinite-dimensional spaces will be offered.

Sequences and series constitute the training ground in which the notions of convergence and limit are first discussed. To better exploit the general understanding of topological and metric spaces built in the previous chapters, we will organise their study in a hierarchical fashion. We will start discussing purely topological properties of sequences, like the definition of convergence itself; the uniqueness of the limit and the limit of subsequences. Then, we will discover what is added if a metric structure is assumed on the space in which the sequence is defined. We will review notions like the boundedness of converging sequences; the existence of the limit for Cauchy sequences and complete metric spaces. A series of results specific to \(\mathbb {R}\) will follow. Many of them derive from the complete order relation defined on this set. I will introduce the notion of upper and lower limits and the (partial) ordering of infinite and infinitesimal sequences. We will discover how to “average” sequences and review several convergence criteria for specific types of sequences. A brief review of the main results in normed linear spaces, essentially \(\mathbb {R}^n\), is also presented. The discussion of series and the convergence of their associated partial sums is presented in \(\mathbb {R}\) because little is gained from considering more general structures. We will deal with the important harmonic and power series. I will discuss the difficulties inherent in dealing with the notion of series and illustrate the most common criteria for deciding on their convergence. Finally, we will explore the notion of sequences of functions and the properties of uniform convergence.

This chapter studies the real-valued functions of one real variable. We start by reviewing what we learnt about the limit of functions and continuity in the previous chapters, adding some results specific to real functions. Then we will introduce the notions of differential and derivative. The technical aspects of computing the derivative of some widely used functions will be covered in a series of examples. More generally, the main focus will be on how to effectively apply the notion of derivative to the study of the local and, to a lesser extent, global behaviour of functions.

In this chapter, we study multivariate functions defined over a subset of the multidimensional real space \(\mathbb {R}^n\) and having values in \(\mathbb {R}^m\). We discuss the differential of multivariate functions and see how this tool can be applied to analyse and characterise their local behaviour. We will review the powerful results of implicit and inverse function theory, and we will apply these results to the solution of constrained maximisation problems.

This chapter begins by introducing the Riemann integral of a bounded function on a closed interval and listing its properties. We study for which classes of functions the integral does actually exist and how to extend its definition to unbounded intervals and unbounded functions. Next, the fundamental relation between integration and derivation is exposed, together with a series of results that are of common use in applications. Finally, we will review the notion of Stieltjes measure and Stieltjes integral, discussing sufficient conditions for its existence and its relation with the Riemann integral.

This chapter presents a formal introduction to measure theory. We cover several related, but different, problems. The first problem is the extension of the notion of integration of real-valued functions from the real line to a larger class of spaces. This problem will be addressed by the theory of measurable spaces and Lebesgue integrals. In particular, the theory of Lebesgue integrals will be used to introduce the notion of multiple integrals, that is, the integral of functions of many variables, in a very natural way. There is, however, a price to pay: we have first to build and discuss the Lebesgue measure on the real numbers. This is a rather tedious exercise, but as a final prize, in addition to the possibility of extending the theory of integration to higher-dimensional spaces, one also gains a better understanding of the Riemann integral and its limitations. The material in this chapter is also important for a more formal study of probability theory, statistics, and stochastic processes. A satisfactory coverage of this topic is outside the scope of the present book, but I will introduce basic notions and useful notations. The discussion of the different topics includes various examples and counterexamples aimed at clarifying the results and dispelling myths. The examples will also serve the purpose of adding some intuition to the rather abstract results of formal measure theory.