Fundamentals of Advanced Mathematics 2

This book is the second in a trilogy of texts covering a wide selection of topics in advanced mathematics, particularly topics that find relevance to engineering and systems theory. The first volume dealt with algebra; our review of that text gives a more detailed description. This volume starts with algebra but soon turns to analysis and topology.

In more detail: the first of the five chapters in this text is on Galois theory, and covers not only the familiar algebraic theory but also its differential version, which has considerable parallels with the algebraic theory but which, basically, addresses solvability of differential equations rather than polynomial ones.

Analysis is pursued in the next several chapters as well. Chapter 2 is on topology. Assuming a basic knowledge of metric spaces, the author develops topology in this chapter at a fairly general level, including the theory of uniform spaces, filters and nets, and topological algebra. The next chapter is on functional analysis, again done at the fairly general level of topological vector spaces rather than Banach spaces.

The next chapter covers several topics. It begins with measure and integration theory, discussing both abstract and Radon measures (some familiarity with Lebesgue theory is assumed), proceeding from here to the theory of analytic functions (the functions here map into complex Banach spaces), and culminates in a discussion of function spaces and distributions.

The fifth and final chapter of the text discusses sheaves. This chapter culminates in a topic I had never seen before, the theory of hyperfunctions, which are then applied to systems of linear differential equations.

Galois theory, topology, topological vector spaces, measure theory, complex analysis, and sheaves may seem like an eclectic selection of topics, but the author explains the reason for their inclusion: the central theme of this book is solution of equations, with both “equations” and “solutions” being interpreted in a fairly broad way. The equations considered are both polynomial and differential, and the solutions involved include “generalized solutions” — both distributions and hyperfunctions. Chapters 2 and 3 are largely prefatory to the fourth and fifth chapters, which address, respectively, the two kinds of generalized solutions just mentioned.

As in the first volume of this trilogy, Bourlès writes very concisely and covers a lot of material in a relatively short amount of text. For example, in the space of about 20 pages, he proceeds from the definition of “topological space” to a statement (and proof, using filters) of Tychonoff’s theorem for general products. Sometimes, I thought he was a bit more concise than he needed to be. When defining a topology \(\mathcal{T}\) on a space \(X\), for example, Bourlès omits the requirement that \(\mathcal{T}\) contain the empty set and \(X\) itself; he states that this follows from the other two assumptions. This is true, but only if \(\mathcal{T}\) is nonempty, an assumption that he does not make.

There are no formal end-of-section exercise sets, but the author frequently, in the middle of textual discussions, points out that the proof of some assertion is left as an exercise. Some proofs, in fact, are omitted entirely (as is the case, for example, in the differential Galois theory portion of the book). However, when this happens, the author generally provides a helpful reference to a book where a proof can be found.

The English is at times a bit idiosyncratic. Referring to the difference between topological and metric spaces, for example, Bourlès writes: “Metric spaces allow us not only to say whether the point \(x’\) is near to the point \(x\), but also whether the points \(x\) and \(x’\) are close to each other”. It seems to me that most readers in this country will view “near” and “close to” as synonyms, making this sentence a bit confusing.

As with the first volume, I’m not sure I see this book having extensive use as a textbook (not only because of the succinct exposition, but because the array of topics covered doesn’t match up with any standard course at the graduate level). However, also as with the first edition, the considerable amount of material included here and the efficient, concise way in which it is presented makes this book valuable as a reference.

Mark Hunacek (mhunacek@iastate.edu) teaches mathematics at Iowa State University.