1.1. Publisher's Summary.

Foreword: The book, "Complex Analysis in Locally Convex Spaces," introduces modern infinite dimensional complex analysis or infinite dimensional holomorphy, as it is commonly called, for the graduate student and research mathematician. The book presents a reasonably comprehensive view of the topological nature of the theory, but has neglected largely the geometric, algebraic, and differential aspects. All of these aspects are equally important and interrelated and a proper appreciation of the portion of the theory outlined in the book is not possible without an overall view of these other topics. The main prerequisite for reading the book is a familiarity with the elements of functional analysis. An acquaintance with several complex variable theories is useful but not essential.

Chapter 1. Polynomials on Locally Convex Topological Vector Spaces: There are two standard methods of introducing polynomials: by using tensor products or by using n linear mappings. The chapter discusses polynomial mappings among vector spaces over $\mathbb{C}; \mathbb{C}, \mathbb{R}, \mathbb{N},$ and $\mathbb{Z}$ denote the complex numbers, real numbers, natural numbers, and integers, respectively. The chapter discusses various kinds of continuous polynomial mappings among locally convex spaces. It also reviews vector spaces of polynomials with locally convex topologies and introduces a duality theory. The chapter presents the polarisation formula, which has many applications. This formula allows comparing the moduli of symmetric n-linear forms and polynomials on certain sets. Because it is usually easier to compute the modulus of an n linear form, the modulus of a polynomial proves useful.

Chapter 2. Holomorphic Mappings between Locally Convex Spaces: This chapter discusses the various definitions of holomorphic mappings among locally convex spaces. The compact open topology and the $\tau_{\omega}$ topology easily extend from polynomials to holomorphic functions on locally convex spaces, but the strong topology does not generalise in a suitable fashion. While the $\tau_{\omega}$ topology plays an important role in the study, it does not generally have good topological properties. The chapter introduces the $\tau_{\delta}$ topology, which may be described as the topology supported by the countable open covers. It also presents the elementary properties of topologies and provides several simple examples. The chapter describes the germs of holomorphic functions, which play a role in the duality theory.

Chapter 3. Holomorphic Functions on Balanced Sets: Balanced open sets are the natural domain of the convergence of the Taylor series expansion at the origin for holomorphic functions. This chapter presents all holomorphic functions on a balanced open set $U$ and demonstrates that the Taylor series expansions lead to a topological decomposition of $H(U)$ for different topologies. This decomposition is used to deduce topological properties of $H(U)$ and to extend results about the spaces of homogeneous polynomials to the spaces of holomorphic functions. The main tools are associated topologies and Schauder decompositions. The theory of holomorphic functions on balanced open sets may be regarded as a local theory. In linear functional analysis, there are two properties - those preserved under locally convex inductive limits and those preserved under projective limits. Many other properties arise as a combination of properties from these two types, for example, reflexivity is defined as semi-reflexivity plus infrabarrelledness and the combined property is not preserved under either kind of limit.

Chapter 4. Holomorphic Functions on Banach Spaces: Banach spaces and nuclear spaces play an important role in linear functional analysis and in classical analysis by a way of application. This chapter presents the study of holomorphic mappings among Banach spaces. It discusses holomorphic functions on nuclear spaces. There is a rich interaction between the theory of holomorphic functions and the geometry of Banach spaces. The geometry of Banach spaces refers to the study of geometric properties of the unit ball, such as smoothness, the existence of extreme points, dentability, uniform convexity, and sequential compactness. The theory of holomorphic functions of one or several complex variable contains a number of interesting and useful equalities and inequalities and it is natural to extend these to infinitely many variables. The chapter presents three well-known results from the theory of one complex variable: Schwarz's lemma, the maximum modulus theorem, and the Cauchy-Hadamard formula.

Chapter 5. Holomorphic Functions on Nuclear Spaces with a Basis: Nuclear holomorphic functions and nuclear spaces enter into the general theory of infinite dimensional holomorphy in a natural way. This chapter discusses the examination of holomorphic functions on fully nuclear spaces. The basis provides a coordinate system, and hence the refinements of a number of techniques are used. The chapter discusses the way to remove the basis hypothesis in certain cases. The existence of a basis also allows defining infinite dimensional analogues of Reinhardt domains and polydiscs. The chapter presents linear and geometric properties of certain classes of nuclear spaces with a basis. It shows that $(H(U), \tau_{0})$ has an absolute basis whenever $U$ is an open polydisc in a fully nuclear space with a basis. A duality theorem is presented in the chapter, which clarifies the relationship among the different topologies on $H(U)$ and places a number of counterexample in perspective.

Chapter 6. Germs, Surjective Limits,$\epsilon$-Products and Power Series Spaces: This chapter discusses the general theory and presents three methods - the $\tau_{\pi}$ topology, surjective limits, and $\epsilon$-products - for studying the relationship among the topologies $\tau_{0}, \tau_{\omega}$, and $\tau_{\delta}$. The $\tau_{\pi}$ topology aims at removing geometric restrictions on the domain; surjective limits are used to generate spaces of holomorphic interest and vector-valued functions can be satisfied by using $\epsilon$-products. The chapter also discusses other problems of general interest, such as the representation of analytic functionals and the completeness of $H(K). K$ is a compact subset of a locally convex space. The positive and negative results obtained show that these are indeed complex issues and not unrelated to one another. $H(K)$ is regular when $K$ is a compact subset of a Fréchet space and it is complete when $K$ is balanced. Holomorphic functions on the strong duals of certain power series spaces are also presented in the chapter.

Appendix I. Further Developments in Infinite Dimensional Holomorphy: This chapter provides a brief survey of some research currently being developed within infinite dimensional holomorphy. The topics discussed in the chapter emphasise the algebraic, geometric, and differential, rather than the topological aspects of the theory. A set of conditions on a domain in a locally convex space is presented in the chapter. It discusses the sheaf theory and sheaf cohomology, which plays an important role in several complex variable theories, and it is probable that the same remark will eventually apply to infinite dimensional holomorphy. Banach algebra considerations motivated the first examples of sheaf cohomology with values in a sheaf of holomorphic germs in infinitely many variables. The chapter also discusses convolution operators and partial differential operators on the spaces of holomorphic functions over locally convex spaces.

Appendix II. Definitions and Results from Functional Analysis, Several Complex Variables and Topology: This chapter presents a list of the definitions and results that are either frequently used or quoted without proof in the book, "Complex Analysis in Locally Convex Spaces," such as the Hausdorff topological space, the Mackey space, the Schwartz space, and more.

Appendix III. Notes on Some Exercises: This chapter presents several results on finite open topology, abstract space, the Banach space, Hartogs' theorem, locally convex space, and more.

1.2. From the Preface.

The main purpose of this book, based on a course at Universidade Federal do Rio de Janeiro during the summer of 1978. was to provide an introduction to modern infinite dimensional complex analysis, or infinite dimensional holomorphy as it is commonly called, for the graduate student and research mathematician. Since we were more interested in communicating the nature rather than the scope of infinite dimensional complex analysis and since it was clearly impossible to write a comprehensive account of the whole theory for such a short course we were obliged to limit our range and choose to develop a single theme which has made much progress in recent years and which exemplifies the intrinsic nature of the subject, namely the study of locally convex topologies on spaces of holomorphic functions in infinitely many variables.

In retrospect, we feel we have provided a reasonably comprehensive view of the topological nature of the theory, but have neglected to a large extent the geometric, algebraic and differential aspects. All of these aspects are equally important, interrelated and indeed a proper appreciation of the portion of the theory outlined in this book is not possible without an overall view of these other topics. To partially compensate for this deficiency we have written Appendix I in which we outline developments in other areas of infinite dimensional holomorphy.

1.3. Review by: Martin Schottenloher.
Mathematical Reviews MR0640093 (84b:46050).

Complex analysis in locally convex spaces (also called infinite dimensional holomorphy) deals with holomorphic functions or mappings on an open subset of a locally convex complex space E (or on a manifold U modelled on E). The book under review emphasises one main aspect of this theory, namely, the study of various natural locally convex topologies on the space H(U) of holomorphic functions on U. The properties of these topologies are closely related to the geometric, algebraic and analytic aspects of the theory.
...
The book is divided into six chapters. The first two chapters are of an introductory character. Chapter 1 deals with polynomial mappings between two locally convex complex spaces $E$ and $F$, including essentially all the now classical results of Fréchet, Gâteaux, Mazur, Orlicz, Michal, Taylor and others.... Chapters 4 and 5 are devoted to the two most important classes of spaces, the Banach spaces and the nuclear spaces. ... In the last chapter some more tools in studying holomorphic functions and mappings are introduced ...

Because of the clear, complete and careful presentation the book is certainly of great value for those working in the field, and it can be expected that it will stimulate further research. Moreover, the book makes accessible many of the recent results in infinite-dimensional holomorphy to a wider audience.

2.1. From the Publisher.

The Schwarz lemma is among the simplest results in complex analysis that capture the rigidity of holomorphic functions. This self-contained volume provides a thorough overview of the subject; it assumes no knowledge of intrinsic metrics and aims for the main results, introducing notation, secondary concepts, and techniques as necessary. Suitable for advanced undergraduates and graduate students of mathematics, the two-part treatment covers basic theory and applications.

Starting with an exploration of the subject in terms of holomorphic and subharmonic functions, the treatment proves a Schwarz lemma for plurisubharmonic functions and discusses the basic properties of the Poincaré distance and the Schwarz-Pick systems of pseudodistances. Additional topics include hyperbolic manifolds, special domains, pseudometrics defined using the (complex) Green function, holomorphic curvature, and the algebraic metric of Harris. The second part explores fixed point theorems and the analytic Radon-Nikodym property.

2.2. From the Introduction.

In this course we discuss intrinsic metrics and distances on complex manifolds. The underlying areas are complex analysis (of one, several, and infinitely many variables), functional analysis (Banach space theory), differential geometry, and potential theory. We assume some familiarity with complex analysis and Banach space theory but assume no knowledge of differential geometry or potential theory. It is our hope that this course will lead to a better understanding of the interconnections between some of the many different aspects of complex analysis and stimulate further research.

2.3. Review by: Teddy J Suffridge.
Mathematical Reviews MR1033739 (91f:46064).

The Schwarz lemma is a basic and very useful result in the theory of analytic functions of one complex variable. Due largely to its wide applicability to analytic functions in the plane, various generalisations have been established - e.g. for subharmonic functions and in higher dimensions - that have proved to be interesting and illuminating as well as extremely useful. In this interesting monograph, the author ably presents some of the beauty, the wide scope and the utility of the Schwarz lemma.

3.1. From the Preface.

This book was initially based all a short course of 20 lectures, given to second year students at University College Dublin during the autumn of 1992. Later, two chapters on Integration theory were added to improve the balance between differential and integral calculus. The students had completed a one-year course on differential and integral calculus for real valued functions of one real variable - this is the prerequisite for reading this book - and this course was designed as an introduction to the calculus of several variables.

My initial motivation for writing this book was to provide my own students with a friendly set of notes that they could read in their entirety. As the book took shape, I realised that I was addressing the subject in a manner somewhat different from the standard texts on several variable calculus. It is difficult to explain precisely why this occurred. Nevertheless, an attempted explanation may also help you, the reader, in your approach and I will try to give a partial one.

Research mathematicians typically spend their working lives doing research, learning new mathematics and teaching. They teach themselves new mathematics mainly to further their own research. Yet, often their own way of learning mathematics is the complete opposite of the way they present mathematics to their students. On approaching a new area of mathematics, the research mathematician is usually looking for some result (or technique). He or she will generally not know precisely what is being sought and lives in hope that by searching, often backwards and forwards through a text, the required result will somehow be recognised. The search through the literature will neither be random nor logical, but will be based on accumulated experience and intuition. Once the objective has been identified the research mathematician works backwards to satisfy professional standards for a precise meaning of the terms involved and the context in which the result may be applied. Finally, and this depends on many things, the research mathematician may even decide to satisfy a need for certainty and will then work through the background proofs. Thus the mathematician, when doing research, behaves like a detective and in fact there is no alternative since the plot is not revealed until the story is almost over. Nevertheless, with students we first reveal the climax (theorem), then the evidence (proof) and finally the intuition (explanation and examples). This robs the subject of its excitement and does not use the students' own intuition and experience. I have tried to approach the material of these lectures as a research mathematician approaches research: full of doubt, more intuitively than logically, somewhat imprecise about where we may be going, but with a general objective in mind, moving backwards and forwards, trying simple cases, using various tricks that have previously proved useful, experimenting and eventually arriving at something of interest. Having obtained useful results intuitively, I have returned to justify them mathematically. At this stage the reasoning behind the proofs is often more acceptable end the proofs themselves become all integral part of a unified process by adding to our understanding of the applications, by showing the usefulness of earlier theoretical results and by suggesting further developments. Of course, I have not fully succeeded in this attempt, but feel nevertheless that I have gone some way in this direction. I believe that this is almost the only way to learn mathematics and that most students are trying to follow this approach.

Although the calculus of several variables is often presented as a fully mature subject in its own right, it is clear that most of the concepts are the natural evolution of attempting to imitate the one-dimensional theory and I have tried to follow this approach in my presentation. The restriction to functions of two variables simplifies the notation and at the same time introduces most of the main concepts that arise in higher dimensions. I believe that a clear understanding of the two-variables case is a suitable introduction to the higher dimensional situation. I have tried to be both rigorous and self-contained and so have clearly marked out assumptions made and discussed the significance of results used without proof.

We discuss all possible functions which involve two variables and so look at functions from $\mathbb{R}^{2} \rightarrow \mathbb{R}, \mathbb{R} \rightarrow \mathbb{R}^{2}$ and $\mathbb{R}^{2} \rightarrow \mathbb{R}^{2}$. This provides a basic introduction to three subjects, i.e., calculus of several variables, differential geometry and complex analysis.

In the first 12 chapters we discuss maxima and minima of functions of two variables on both open sets and level curves. Second order derivatives and the Hessian are used to find extremal values on open sets, while the method of Lagrange multipliers is developed for level curves. In the process we introduce partial derivatives, directional derivatives, the gradient, critical points, tangent planes, the normal line and the chain rule and also discuss regularity conditions such as continuity of functions and their partial derivatives and the relationship between differentiation and approximation. In Chapters 13 to 16 we investigate the curvature of plane curves. Chapters 18 to 22 are devoted to integration theory on $\mathbb{R}^{2}$. We study Fubini's theorem (on the change of order of integration), line and area integrals and connect them using Green's theorem. In Chapter 17 we introduce holomorphic (or $\mathbb{C}$-differentiable) functions and using approximation methods derive the Cauchy-Riemann equations. This introduction to complex analysis is augmented in the final chapter where Green's theorem is combined with the Cauchy-Riemann equations to prove Cauchy's theorem. Partial derivatives enter into and play an import ant role in every topic discussed.

As life is not simple, many things depend on more than one variable and it is thus not surprising that the methods developed in this book are widely used in the physical sciences, economics, statistics and engineering. We mention some of these applications and give a number of examples in the text.

Anyone interested in several variable calculus will profit from reading this book. Students suddenly exposed to the multidimensional situation in its full generality will find a gentle introduction here. Students of engineering, economics and science who ask simple but fundamental questions will find some answers and, perhaps, even more questions here.

This book may be used fully or partially as the basis for a course in which a lecturer has the option of inserting extra material and developing more fully certain topics. Alternatively it can be used as supplementary reading for courses on advanced calculus or for self study. The material covered in each chapter can be presented in approximately 60 minutes, although in some of my lectures I was not able to cover fully all examples in the allocated time, and for aesthetical and mathematical reasons I have, in writing up these lectures, sometimes ignored the time frame imposed by the classroom.

4.1. From the Preface.

The importance assigned to accuracy in basic mathematics courses has, initially, a useful disciplinary purpose but can, unintentionally, hinder progress if it fosters the belief that exactness is all that makes mathematics what it is. Multivariate calculus occupies a pivotal position in undergraduate mathematics programmes in providing students with the opportunity to outgrow this narrow viewpoint and to develop a flexible, intuitive and independent vision of mathematics. This possibility arises from the extensive nature of the subject.

Multivariate calculus links together in a non-trivial way, perhaps for the first time in a student's experience, four important subject areas: analysis, linear algebra, geometry and differential calculus. Important features of the subject are reflected in the variety of alternative titles we could have chosen, e.g. 'Advanced calculus', 'Vector calculus', 'Multivariate calculus', 'Vector geometry', 'Curves and surfaces' and 'Introduction to differential geometry'. Each of these titles partially reflects our interest but it is more illuminating to say that here we study differentiable functions, i.e. functions which enjoy a good local approximation by linear functions.

The main emphasis of our presentation is on understanding the underlying fundamental principles. These are discussed at length, carefully examined in simple familiar situations and tested in technically demanding examples. This leads to a structured and systematic approach of manageable proportions which gives shape and coherence to the subject and results in a comprehensive and unified exposition.

4.2. Review by: Peter Shiu.
The Mathematical Gazette 82 (495) (1998), 536-537.

The main difficulty experienced by university students of mathematics in the study of multivariate calculus lies in their reluctance to master the necessary techniques. It has to be said that this is not due to laziness on their part, but because they often lack a true understanding and appreciation of the concepts involved. Besides familiarity with one-variable calculus and linear algebra, students will need to have some basic knowledge of analysis, such as the argument that a continuous real-valued function on a compact set attains a maximum and a minimum.

In order to enable readers of such a text to apply the theory with confidence the theorems must be formulated properly. At the same time the statements of those theorems should not be so wide ranging that the majority of the readers find them formidable or even intimidating, and there is little point in giving rigorous proofs. In the present text, the author has taken the sensible approach of giving an adequate formulation of the theories, backed up by geometric insight and accompanied by reasonable explanations. Instead of burdening the reader with tedious proofs, intuitive arguments are applied. Equally importantly, suitable examples to illustrate the uses and limitations of multivariate calculus are given at each stage of development.

There are 18 chapters covering three broad areas: Differential Calculus on Open Sets and Surfaces; Integration Theory, which deals particularly with the key concepts of parametrisations and orientated surfaces, leading to the divergence theorems; Geometry of Curves and Surfaces, which includes the use of vector-valued differentiation to obtain the Frenet-Serret equations, together with chapters on Gaussian curvature and geodesic curvature. Each chapter opens with a short summary of its content, and all the exercises have solutions. There is also a useful table of parametrisations with various coordinate systems. Students should find this clearly written text valuable both for learning and for reference.

5.1. Review by: José Bonet.
Mathematical Reviews MR1705327 (2001a:46043).

This book presents an exhaustive exposition of the main developments in infinite-dimensional holomorphy in the last two decades. As the author writes in the first lines of his preface, infinite-dimensional holomorphy is the study of holomorphic and analytic functions over complex topological vector spaces. The definitive step in the creation of complex infinite-dimensional analysis was taken by Volterra in 1887. Hadamard realised the importance of Volterra's work, and he had a strong influence on Fréchet and Gateaux. H von Koch introduced a monomial approach to holomorphy on infinite-dimensional polydiscs which was developed by Hilbert in 1909. The current definition of holomorphic mapping was given independently by Graves and A E Taylor in the late 30's. Analyticity played a relevant role in operator and spectral theory. M A Zorn made a number of important contributions in the mid-40's. The topological vector structure of the space of holomorphic functions defined on an open subset of $\mathbb{C}^{n}$ was studied by Grothendieck and Köthe (1953) and Martineau (1966). Nachbin, in a series of articles between 1966 and 1970, introduced and studied several topologies on the space of holomorphic functions on an infinite-dimensional space which are analysed in Dineen's book.

The author had written a book on the same subject before [Complex analysis in locally convex spaces, North-Holland, Amsterdam, 1981]. Like that volume, the present one also emphasises properties of different topologies defined on spaces of holomorphic functions, and studies the relations between them. However, in the last 20 years considerable progress has been made in the study of polynomials and tensor products. This development is clearly reflected in the book. Polynomials are mainly a tool, but the first two chapters are a self-contained study of them, their duality, geometry and topologies. Here tensor products play an important role. These two chapters reflect that polynomials are nowadays an independent area of research within (multi-)linear functional analysis.

The present book is independent of the previous one. The latter is not necessary to understand the new one, which is self-contained. The central theme of the book is the space $H(U)$ of holomorphic functions on an open subset $U$ of a complex locally convex space $E$ and the relationships among three topologies on this space ...
...
The book reflects very well the interaction between (multi-)linear functional analysis and complex-analytic function theory. Despite the amount of information and the many references to recent work, the book is not only a reference tool. It is very readable and will be useful to readers who are interested not only in polynomials or infinite holomorphy, but also in Banach spaces, locally convex spaces, topological tensor products, complex function theory, several complex variables, topological algebras, etc. This important book culminates years of exhaustive research, and it will be indispensable for present and future researchers in this and related areas.

5.2. Review by: Ignacio Zalduendo.
Bulletin of the Irish Mathematical Society 44 (2000), 94-97.

One might attempt to define infinite dimensional holomorphy as the study of differentiable non-linear functions on infinite dimensional (usually complex) topological vector spaces. But then this immediately strikes us as so very remote from the growing field that it now is. The present picture of infinite dimensional holomorphy intersects with many branches of mathematics: differential geometry, Jordan algebras, Lie groups, operator theory, logic. It is an area of vigorous research, with connections to many other areas.

It is generally agreed that infinite dimensional holomorphy begins before the XXth century, and the works usually cited as the first in the field are often also cited as being the roots of Functional Analysis. Volterra uses the Taylor series expansion of a real-valued analytic function on $\mathbb{C}[a, b]$ as early as 1887, von Koch and Hilbert use monomial expansions converging on polydiscs around the turn of the century. Fréchet (1909) defines and uses real polynomials on $\mathbb{R}^{N}$ and $\mathbb{C}[a, b]$, and Gateaux (1912) defines complex holomorphic functions and proves a Cauchy integral formula and Cauchy inequalities. These first advances were on concrete spaces. Functional Analysis was in its infancy, with the definition of normed space still years away. The Polish school was interested in the non-linear: one can find still unsolved problems on infinite variable polynomials in The Scottish Book, mostly by Mazur and Orlicz, and Banach himself promised 'another volume' regarding analyticity would follow his Théorie des Opérations Linéaires. But this was not to be.

The second period of the development of infinite holomorphy starts in America in the 1930's, with Michal and his students. It is a period of unification and clarification of the theory. The relationship between $k$-homogeneous polynomial and $k$-linear mappings is now established, the polarisation formula appears. Taylor series now converge on balanced domains of abstract spaces, and much of the finite-variable theory is generalised. In the 1940's and 1950's, tensor products make their appearance, as well as holomorphy on general locally convex spaces. But linear Functional Analysis overshadows the non-linear in this era, and progress is relatively slow, except in the grey area between the linear and the non-linear: the bilinear and its relation with duality and with the product in topological algebras.

The third era begins in the mid 1960's with Nachbin and his students. Infinite holomorphy comes of age, now it's what doesn't happen in finite dimensions that counts. Holomorphic functions of diverse types (integral, nuclear, weakly continuous) appear, spaces of holomorphic functions have competing topologies, $\tau_{0}, \tau_{\omega}, \tau_{\delta}$, which may or may not coincide. Their duals are studied. Preduals are searched for, as are decompositions into spaces of homogeneous polynomials. Holomorphic automorphisms in infinite dimensions are studied, together with bounded symmetric domains and Jordan triple systems. The Levi problem, domains of holomorphy, plurisubharmonicity, germs of holomorphic functions all make their necessary appearance. Many open problems remain and are being actively studied today. In the mid 1980's polynomials and spaces of homogeneous polynomials began to call attention for their own sake and, fuelled by important advances in Banach space theory, now constitute a highly active field of study. This book appears then, at a time when it is very necessary, for not only has the subject grown, but its focus has changed since the 1980's when most infinite holomorphy texts were published.

Formally, the author assumes only basic knowledge of one complex variable and some experience with Banach space theory. The reader who approaches the book with that background alone will find many sections are hard but worthwhile reading. But the expert will also find much to learn from this book. As the author says in the preface, he has 'tried to maintain a delicate balance between ... self-contained introduction for the non-expert and to provide a comprehensive summary for the expert'. He has succeeded admirably in creating a book that admits several different readings; a book to come back to.

The unifying theme is the relationship in $H(U)$ (where $U$ is an open subset of a locally convex space $E$) between the topologies $\tau_{0}, \tau_{\omega}$ and $\tau_{\delta}$; and the study of how this relationship is affected by properties of $U$ and of the underlying locally convex space $E$. The theory of bounded symmetric domains and Jordan triples is not included. The book begins with polynomials in both the multilinear and tensor product presentations. Topologies in spaces of polynomials are compared, and geometric properties of these spaces are discussed. Chapter 2 presents duality theory and reflexivity for spaces of polynomials. This is the opportunity to introduce several different classes of polynomials: approximable, nuclear, integral, weakly sequentially continuous, as well as the properties of the underlying space necessary to obtain positive results: the approximation property, nuclearity, the Radon-Nikodym property, non-containment of 1. Gateaux and Fréchet holomorphic mappings make their appearance in Chapter 3, together with the ways of approximating them, monomial and Taylor series expansions. The different topologies on spaces of holomorphic mappings are introduced and their properties studied. In Chapter 4, the approximation of holomorphic functions are applied to the comparison of topologies on $H(U)$ for a balanced open subset $U$ of a Fréchet space. The comparison of topologies is then carried out in Chapter 5 for arbitrary open subsets $U$. This entails a discussion of Riemann domains over locally convex spaces, and of the Levi problem and Cartan-Thullen theorem. The book ends with the study of holomorphic extensions from dense subspaces and from closed subspaces, and with a discussion of the space of holomorphic functions of bounded type as an algebra, and the study of its spectrum.

The pace of the book is a sustained one, but the exposition is fresh and full of short excursions into other topics as the need for these arises. Thus one finds a nice discussion of the (BB)-property, a short introduction to the theory of locally convex spaces, and an excursion to the local theory of Banach spaces. Further on, the role played by quojections, the need for DFN spaces and nuclearity. An introduction to Schauder decompositions, a discussion of Riemann domains, another of Arens-regularity. These are the threads which tie the infinite holomorphy core of the book to the rest of Analysis, giving it its place in mathematics and ultimately, its beauty.

This is a scholarly work. The author has made a fruitful effort at simplifying and unifying techniques and providing new proofs. Each chapter is accompanied by many exercises - over 250 in all - most of which are commented in an appendix, and by extensive and carefully researched historical notes. The list of references has over 850 entries. A very complete book. And at a very reasonable price, too.

But perhaps the most valuable thing that the author has put into this book is his insight and his perspective. His views on why such a property will or will not play a significant role in the future, and how the subject is developing are sprinkled liberally throughout the text. This is a great attraction for the reader, who will come back for more, and in each reading, will find it. We mathematicians take pride in the timelessness of our work: mathematics - more than any other human endeavour - is forever. I sometimes wonder, then, why mathematics books grow old so fast. A mathematical classic is any book older than its reader. In my bookcase this definition includes only a venerable handful: Landau, Hardy-Littlewood-Polya, Kelley, Banach, Zygmund. Dineen will be - I am sure - a classic.

6.1. From the Publisher.

Multivariate calculus, as traditionally presented, can overwhelm students who approach it directly from a one-variable calculus background. There is another way-a highly engaging way that does not neglect readers' own intuition, experience, and excitement. One that presents the fundamentals of the subject in a two-variable context and was set forth in the popular first edition of Functions of Two Variables.

The second edition goes even further toward a treatment that is at once gentle but rigorous, atypical yet logical, and ultimately an ideal introduction to a subject important to careers both within and outside of mathematics. The author's style remains informal and his approach problem-oriented. He takes care to motivate concepts prior to their introduction and to justify them afterwards, to explain the use and abuse of notation and the scope of the techniques developed.

Functions of Two Variables, Second Edition includes a new section on tangent lines, more emphasis on the chain rule, a rearrangement of several chapters, refined examples, and more exercises. It maintains a balance between intuition, explanation, methodology, and justification, enhanced by diagrams, heuristic comments, examples, exercises, and proofs.

6.2. Review by: kz.
European Mathematical Society Newsletter 46 (December 2002), 37.

While standard textbooks aim at a more mathematically oriented audience, this little book is a gentle introduction to multivariate calculus in a two-variable context. It is accessible to students with a diverse and modest background and interest in mathematics, science or engineering. Unlike many modern presentations, this book begins with the particular and works its way to the more general, helping the student to develop an intuitive feeling for the subject.

After an elementary treatment of basic concepts, the subsequent chapters carefully introduce Lagrange's multipliers, the chain rule, curvature, quadratic approximation, etc. The author has saturated the text with illustrations and with many interesting and explicitly calculated examples; there are also many inspiring exercises. The work ends with double integrals and Green's theorem. This edition includes a rearrangement of several chapters, new examples and exercises. The whole text is nicely written, and can be strongly recommended as an excellent and comprehensive source, suitable for self-study or classroom use at the undergraduate level. For students demanding motivation, its study will be a rewarding experience.

7.1. From the Preface.

This second edition gave me the opportunity to streamline some arguments, to correct a number of errors and misprints, to provide additional exercises and to add some further details to the solutions.

7.2. Review by: Peter Saunders.
The Mathematical Gazette 86 (505) (2002), 180-181.

Most readers will be familiar with the sort of American college textbook that is called 'Calculus and Geometry' or something like that. These are large books designed for freshman mathematics courses and they all cover much the same material in much the same way. They generally include an introduction to the calculus of more than one variable, vector algebra and calculus, and three dimensional analytic geometry, or if they do not, there may be a more advanced sequel that does.

If that's what you want, this book, despite its title, isn't what you're looing for. True, Dineen deals with many of the topics you would find in a typical text on advanced calculus but he does so in a much more sophisticated way. You only have to look at the definition (on page 3!) of a continuous function to see the difference. And by geometry, he means an introduction to differential geometry, the geometry of curves and surfaces.

Dineen writes in the preface that all you need to read this book is a knowledge of one-variable differential and integral calculus, partial differentiation, and an elementary understanding of matrices. Strictly speaking, this is so. All the new terms are defined as they appear and all the necessary theorems are stated, though not always proven. But the reader is plunged immediately into an advanced notation and terminology, and someone with such a limited mathematical background is going to have real trouble understanding what is going on, even if he reads Chapter 1 several times, as Dineen suggests.

Incidentally, in view of the demands that Dineen makes on the reader in other ways, I was surprised that he generally works out integrals in great detail: even something as straightforward as $\int _{0}^{2} r√(1+4r^{2}) dr$ takes several lines. I found myself wondering whether this was just an idiosyncrasy, or whether he finds his students are held back by a lack of facility in carrying out simple calculations - in which case he is not alone!

I wouldn't recommend this book as an introduction to the calculus of more than one variable. In my experience, students find it hard enough to come to grips with the subject using methods and terminology with which they are familiar; I'd be very reluctant to try to introduce them to it in a language they aren't accustomed to.

The book is, however, very clearly written, and there are lots of useful diagrams and worked examples. The approach, even though it may not be suitable for beginners, is one that students who are going on will want to learn sooner or later. Once they have learned enough multivariable calculus to be comfortable with it, this book offers an excellent opportunity for them to learn the more modern approach and to use it right from the start when they move on to differential geometry.

8.1. From the Publisher.

The use of the Black-Scholes model and formula is pervasive in financial markets. There are very few undergraduate textbooks available on the subject and, until now, almost none written by mathematicians. Based on a course given by the author, the goal of this book is to introduce advanced undergraduates and beginning graduate students studying the mathematics of finance to the Black-Scholes formula. The author uses a first-principles approach, developing only the minimum background necessary to justify mathematical concepts and placing mathematical developments in context.

The book skilfully draws the reader toward the art of thinking mathematically and then proceeds to lay the foundations in analysis and probability theory underlying modern financial mathematics. It rigorously reveals the mathematical secrets of topics such as abstract measure theory, conditional expectations, martingales, Wiener processes, the Ito calculus, and other ingredients of the Black-Scholes formula. In explaining these topics, the author uses examples drawn from the universe of finance. The book also contains many exercises, some included to clarify simple points of exposition, others to introduce new ideas and techniques, and a few containing relatively deep mathematical results. With the modest prerequisite of a first course in calculus, the book is suitable for undergraduates and graduate students in mathematics, finance, and economics and can be read, using appropriate selections, at a number of levels.

8.2. From the Preface.

Mathematics occupies a unique place in modern society and education. It cannot be ignored and almost everyone has an opinion on its place and relevance. This has led to problems and questions that will never be solved or answered in a definitive fashion. At third level we have the perennial debate on the mathematics that is suitable for non-mathematics majors and the degree of abstraction with which it should be delivered. We mathematicians are still trusted with this task and our response has varied. Some institutions offer generic mathematics courses to all and sundry, and faculties, such as engineering and business, respond by directing their students to the courses they consider appropriate. In other institutions departments design specific courses for students who are not majoring in mathematics. The response of many departments lies somewhere in between. This can lead to tension between the professional mathematicians' attitude to mathematics and the client faculties' expectations. In the first case non-mathematics majors may find themselves obliged to accept without explanation an approach that is, ill their experience, excessively abstract. In the second. a recipe-driven approach often produces students with skills they have difficulty using outside a limited number of well-defined settings. Some students, however, do arrive, by sheer endurance, at an intuitive feeling for mathematics. Clearly both extremes are unsatisfactory and it is natural to ask if an alternative approach is possible.

It is, and the difficulties to be overcome are not mathematical. The understanding of mathematics that we mathematicians have grown to appreciate and accept, often slowly and unconsciously, is not always shared by non-mathematicians, be they students or colleagues, and the benefits of abstract mathematics are not always obvious to academics from other disciplines. This is not their fault. They have, for the most part, been conditioned to think differently. They accept that mathematics is useful and for this reason are willing to submit their students to our courses. We can - and it is in our own hands, since we teach the courses - show that it is possible to combine abstract mathematics and good technical skills. It is not easy, it is labour intensive. and the benefits are usually not apparent in the short term. It requires patience and some unconditional support that we need to earn from our students and colleagues.

Although this book is appearing as a graduate text in mathematics, it is based on a one-semester undergraduate course given to economics and finance students at University College Dublin. It is the result of an opportunity given to the author to follow an alternative approach by mixing the abstract and the practical. We feel that all students benefited, but some were not convinced that this was indeed the case.

The students had the usual mathematical background, an acquaintance with the techniques of one variable differential and integral calculus and linear algebra. The aim of the course was to provide a mathematical foundation for further studies in financial mathematics, a discipline that has made enormous advances in the last twenty-five years and has been the surprise catalyst in the introduction of certain high-level mathematics courses for non-mathematics majors at universities in recent years. Even though the eventual applications are concrete, the mathematics involved is quite abstract, and as a result business students. who specialise in finance, are today exposed to more demanding mathematics than their fellow students in engineering and science. The students' motivation, background, aspirations and future plans were the constraints under which we operated, and these determined the balance between the choice of topics, the degree of abstraction and pace of the presentation.

In view of its overall importance there was no difficulty in choosing the Black-Scholes formula for pricing a call option as our ultimate goal. This provided a focus for the students' motivation. As the students were not mathematics majors but the majority would have one or two further years of mathematically oriented courses, it seemed appropriate to aim for an understanding that would strengthen their overall mathematical background. This meant it was necessary to initiate the students into what has unfortunately become for many an alien and mysterious subject, modern abstract mathematics. For this approach to take root the security associated with recipe-driven and technique-oriented mathematics has to be replaced by a more mature and intrinsic confidence which accepts a degree of intellectual uncertainty as part of the thinking process. Even with highly motivated students, this requires a gradual introduction to mathematical abstraction, and at the same time it is necessary to remain, for reasons of motivation, in contact with the financial situation.

Probability theory, Lebesgue integration and the Itô calculus are the main ingredients in the Black-Scholes formula, and these rely on set theory, analysis and an axiomatic approach to mathematics. We take, on the financial side, a first principles approach and include only the minimum necessary to justify the introduction of mathematical concepts and place in context mathematical developments. We move slowly initially and provide elementary examples at an early stage. Hopefully, this makes the apparently more difficult mathematics in later chapters more intuitive and obvious. This cultural change explains why we felt it necessary on occasion to digress into non-technical, and even psychological, matters and why we attempted to present mathematics as a living culture with a history and a future. In particular, we tried to explain the importance of properly understanding questions and recognising situations which required justification. This helped motivate, and place in perspective, the need for clear definitions and proofs. For example, in considering the concept of a convergent sequence of reel numbers, on which all stochastic notions of convergence and all theories of integration rely, we begin by assuming an intuitive concept of limit in Chapter 1; in Chapter 3 we define the limit of a bounded increasing sequence of real numbers; in Chapter 4 we define the limit of a sequence of real numbers; in Chapter 6 we use upper and lower limits to characterise limits; in Chapter 9 we use Doob's upcrossing approach to limits; and in Chapter 11 we employ subsequences to obtain an equivalent definition of limit. In all cases the different ways of considering limits of sequences of real numbers are used as an introduction to similar but more advanced concepts in probability theory.

The introduction of peripheral material, the emphasis on simple examples, the repetition of basic principles, and attention to the students' motivation all take time. The real benefits only become apparent later, both to the students and their non-mathematical academic advisors, when they, the students, proceed to mix with other students in mathematically demanding courses.

The main mathematical topics covered in this book, for which we assume no background, are all essentially within probability theory. These are measure theory, expected values, conditional expectation, martingales, stochastic processes, Wiener processes and the Itô integral. We do not claim to give a fully comprehensive treatment, and we presented, even though otherwise tempted, certain results without proof. Readers who have worked their way through this book should be quite capable of following the standard proofs in the literature of The Central Limit Theorem, The Radon-Nikodym Theorem, etc., and we hope they will be motivated to do so. Our self-imposed attempt at self-sufficiency sometimes led to awkward proofs. Although probability theory was the initial focus for our studies, we found as we progressed that more and more analysis was required. Having introduced sequences and continuous functions and proved a number of their basic properties, it did not require much effort to complete the process and present with complete proofs the fundamental properties of continuous and convex functions in Sections 7.2 and 7.6 respectively.

Different groups may benefit from reading this book. Students of financial mathematics at an early, but not too early, stage in their studies could follow, as our students did, Chapters 1-5; Sections 6.1, 6.2, 6.3 and 7.3; the statements of the main results in Sections 6.3, 6.4, and 7.5; and Chapters 8-10. Students of mathematics and statistics interested in analysis and probability theory could follow Chapters 3-7 with the option of two additional topics: the combination of Section 8.2, Chapter 9 and Section 10.3 forming one topic and Chapter 11 the other. Students of mathematics could follow Chapters 3-6 as an introduction to measure theory, while Chapter 11 is, modulo a modest background in probability theory, a self-contained introduction to stochastic integration and the Itô integral. Finally anyone beginning their university studies in mathematics or merely Interested in modern mathematics, from a philosophical or aesthetic point of view, will find Chapters 1-5 accessible, challenging and rewarding.

The exercises played an important role in the course, on which we based this book. Some are easy, others difficult; many are included to clarify simple points; some introduce new ideas and techniques; a few contain deep results; and there is a high probability that some or our solutions are incorrect. However, an hour or two attempting a problem is never a waste of time, and to make sure that this happened these exercises were the focus of our small-group weekly workshops. This is a secret that we mathematicians all too often keep to ourselves. Mathematics is an active discipline, progress cannot be achieved by passive participation, and with sustained active participation progress will be achieved.

8.3. Review by: Christian-Oliver Ewald.
Mathematical Reviews MR2179650 (2006h:60001).

The aim of this nicely written book is to give a mathematically rigorous and profound treatment of the Black-Scholes equation and its solution the Black-Scholes formula for the value of a European option in the standard Black-Scholes model. Nowadays there are plenty of books covering issues from financial mathematics. This is related to the current boom of starting MSc courses in financial mathematics in the UK and other countries as well. The question is, what distinguishes this book from others? The main difference is that it starts basically with no mathematical prerequisites and ends with the Black-Scholes formula, while developing all mathematical theory thoroughly and in detail, on a level not seen in other textbooks. It is possible, indeed, to teach a course in financial mathematics from this book without making use of any accompanying book covering related mathematical issues, such as elementary probability theory or calculus. The book has been used by the author for teaching a one semester course for undergraduate students at the University College Dublin. Teaching in an MSc in financial mathematics myself, I think it is more realistic to use the book for two consecutive one semester courses, or two parallel courses, one covering the mathematical theory and one on the applications in finance. The idea of having one course text for the whole program, covering all mathematical theory as well as the financial applications is very nice. On the other side, I must say that the mathematical aspect of the book dominates the financial aspect, and I have concerns that the students in a typical MSc in financial mathematics program, who have different backgrounds, coming from economics, business, engineering and mathematics, would not be willing to endure the lengthy mathematical exposition. I think that one of the author's aims is to entice the students by the beauty of mathematics, which he illustrates very well. If a course instructor teaching from the book succeeds in this, my guess is that the course will be a great success. On the other side, the instructor faces the risk of getting lost in mathematical details and the students becoming disoriented and losing interest.
...
... it must be clearly said that this is a very nice book. It outshines most of its competitors with regard to its very interesting historical remarks as well as the variety of nice exercises and solutions which the book contains, as well. In my opinion the book lacks a little bit with regard to practical applications, but this aspect may not have been the author's focus, as there are plenty of books on applications.

8.4. Review by: Errol Caby.
Technometrics 50 (2) (2008), 234-235.

This book seeks to provide a rigorous foundation for financial mathematics, an area that, although having very concrete applications, uses quite abstract mathematics. Its aim is to not only present the standard abstract topics, but also explain some of the intuitive ideas that led to them. It thus seeks to avoid on one hand a technique-oriented approach, which tends to be limited to only well established settings, and on the other hand an abstract approach disconnected from applications, where the knowledge gained cannot be easily applied to real-world problems. To achieve this, the author generally discusses new ideas intuitively before treating them abstractly. He also explicitly addresses the art of thinking mathematically and discusses what is required to increase one's mastery of this art.

The book is self-contained (requiring only an acquaintance with the techniques of differential and integral calculus for one variable plus linear algebra) and covers much ground mathematically, but introduces only basic finance. It is organised into 11 chapters with exercises after each chapter and solutions to most of them. In Chapter 1 the author introduces basic monetary concepts and terms. In Chapter 2 he presents simple gambling examples to introduce and illustrate ideas related to risk associated with the growth of money. In Chapters 3-9 and 11, the author gives a self-contained development of abstract probability theory and, while developing the mathematics, looks at increasingly more complex models for share options. Chapter 10 is devoted to the focal point of the book, the Black-Scholes formula for pricing share options. In what follows I take a more detailed look at the material, and conclude with general comments.

The author introduces basic monetary concepts, such as inflation and interest, in Chapter 1. Various types of interest (e.g., simple interest, interest compounded at time intervals of finite length and interest compounded continuously) are discussed. The form of interest covered subsequently in the book is interest compounded continuously. Arbitrage (i.e., risk-free guaranteed profit) an important monetary concept, is introduced by means of a simple example. The last section of Chapter 1 forms a bridge to Chapter 2, which introduces the growth of money in the face of risk. Here such concepts as expected value and fair games are introduced through simple gambling examples. Hedging (i.e., removing the uncertainty associated with outcomes determined by chance) also is introduced. These examples also are used to introduce important principles used later in more complex and abstract settings.

Starting in Chapter 3, the author takes up the development of abstract probability theory. The chapter begins with a general discussion of the process of abstraction and thinking mathematically. Measurable spaces are then introduced as the abstraction of the intuitive ideas on outcomes and events introduced in Chapter 2. In Chapter 4, another component of probability theory, measurable real-valued functions, is added. In Chapter 5, probability measures are added to measurable spaces to give probability spaces. At this point, additional financial terms, such as call option (i.e., an option to purchase shares at some point in the future at a certain price), put option (i.e., an option to sell shares at some point in the future at a certain price), maturity/exercise date, and strike price (i.e., the predetermined price) are introduced. A simple model for pricing share options also is introduced. In this model the share price is assumed to take only one of two values at the maturity date; that is, share price is assumed to be binomial. One of these values is assumed to be larger than the current share price, the other lower. The price of a share option and hedging strategy is then derived for this model. In doing so, the important concept of "risk-neutral probability" (i.e., the probability distribution of share price that makes buying the share a fair game) is introduced. The price of the option is set so that buying a share option is a fair game when the share price has the risk neutral probability distribution at maturity. This principle is applied to a more complex model to get the Black-Scholes formula for pricing options. The author completes this chapter by defining independence, random variables, and stochastic processes, formalising intuitive ideas that were introduced in previous chapters.

Chapter 6 gives a rigorous development of expected value, which was used intuitively in discussing fair games (Chap. 2) and in Chapter 5's call option example. The discussion starts with simple random variables and examines increasingly more general/complex random variables. The author draws an insightful analogy to the problem of finding the perimeter of an object, a problem solved in ancient times. (Once a unit of length was specified, the perimeter of a simple object whose perimeter could be partitioned into a finite number of straight line segments was calculated; the perimeter of a more complex object, such as a circle, was then found by approximating its perimeter with a finite number of line segments and taking the limit as the approximating figure approached the object.) In this analogy we can see that the same mathematical process that was applied to the problem of finding perimeters also was applied to finding expected values. Chapter 7 continues the discussion of integration including results from real analysis, such as properties of continuous functions on closed bounded intervals and the equality of the Riemann and Lebesgue integrals for continuous functions on closed finite intervals (in practice, we evaluate a specific Lebesgue integral by taking the Riemann integral). The central limit theorem for the case of independent, identically distributed random variables is also stated without proof.

Chapter 8 opens with a model for call option pricing that is slightly more complex than that presented in Chapter 5. In addition to considering the share price at the start and maturity dates, here the share price also is considered at the point midway between these two points. For this model, the result at maturity is seen to be conditional on what happened at the intermediate point in time. This is used to motivate a rigorous development of conditional expectation. In treating this topic, an explicit and illustrative construction of conditional expectation is given when the sigma field conditioned on is generated by a countable partition. Chapter 8 closes by examining hedging in a more realistic model. Here share price is considered over a time interval of finite length. The transition to this model is made by partitioning the time interval into small subintervals, applying the approach of the previous models to each of the subintervals and taking the limit as the widths of the subintervals go to zero. This yields a stochastic integral equation that the hedging strategy (a function of share price and time) should satisfy. Martingales are treated in Chapter 9. The fair game condition requires that the discounted share price, the share price after an interest adjustment, and the discounted value of the option be martingales under the risk-neutral probability distribution. Martingale properties of the Wiener process (Brownian motion) and related processes (e.g., exponential Brownian motion) are derived.

In Chapter 10 the author derives the book's main result, the Black-Scholes formula. This formula gives a fair price of call options when share price is modelled by exponential Brownian motion. He derives this formula in two ways: (1) from first principles, where the time interval is partitioned into subintervals of small length, share price is modelled by a binomial random variable on each of these subintervals, and the limit is taken as the widths of the subintervals go to zero, and (2) using a change of measure result. Finally, in Chapter 11 the hedging strategy for the case in which share price is modelled by exponential Brownian motion is derived. To handle the stochastic integral equation that the hedging strategy must satisfy, a rigorous development of the stochastic Riemann and Ito integrals is first given.

In summary, the author has done a very good job presenting the abstract probability theory needed to understand the derivation of the Black-Scholes formula, as well as the intuitive ideas, often simple, behind it. By starting with simple models in which the financial principles used can be clearly seen and then building smoothly toward the more complex model used in the Black-Scholes formula, the author allows the reader to appreciate the financial principles used even in the more complex settings. Furthermore, the exercises included after each chapter help the reader understand the material presented. (As the author points out, mathematics is not a passive endeavour.) Finally, the many historical footnotes often add insight and help bring the mathematics to life.

9.1. From the Publisher.

This book shows that it is possible to provide a fully rigorous treatment of calculus for those planning a career in an area that uses mathematics regularly (e.g., statistics, mathematics, economics, finance, engineering, etc.). It reveals to students on the ways to approach and understand mathematics. It covers efficiently and rigorously the differential and integral calculus, and its foundations in mathematical analysis. It also aims at a comprehensive, efficient, and rigorous treatment by introducing all the concepts succinctly. Experience has shown that this approach, which treats understanding on par with technical ability, has long term benefits for students.

9.2. From the Preface.

This book is based on an introductory first year course given by the author over a number of years to students of Economics and Finance at University College Dublin. This year long course was initially called, One Variable Differential and Integral Calculus, but was later retitled, Introduction to Analysis. Those business students did not come to university to study mathematics but had knowingly enrolled in a programme that required sophisticated mathematics during each year of their undergraduate studies. As the students were familiar with the technical aspects of the calculus I was motivated to help them understand what they knew and I was aware that today's students, particularly those with expectations of a lengthy career in any discipline that routinely applies mathematics, will need the ability to approach unknown areas of mathematics. This skill requires the confidence that accompanies understanding. In this book we show that it is possible to search for understanding while still meeting traditional expectations. Moreover, we found that mathematics as a language with clearly defined concepts and a reliance on deductive arguments had other long term practical benefits for our students.

The road to understanding is quite different to the technique oriented route. It is necessary to start at the very beginning, to make no assumptions, and to carefully and thoroughly examine one or two topics. The choice of topics is unimportant. The students responded reluctantly to this unfamiliar approach but, as their involvement grew, they cautiously put aside previous attitudes and began to develop a confidence that accepts uncertainty. When this was in place, and it was neither immediate nor universal, technical skills were rapidly absorbed. As I put together handwritten notes to explain some chaotic lectures, I realised I was not addressing the usual audience for mathematical textbooks, reluctant teenagers, and a concise new title suggested itself: Introducing Mathematics to Consenting Adults. For a few years typed notes, with this title, were distributed to the students. Here, I have opted for a title that highlights a different aspect of this book. Analysis means separating a thing into its component parts and, by extension, discovering the general principles underlying individual phenomena. The founders and developers of the calculus used this word in their publications, acknowledging their analysis of infinite processes. Since then the word has been used in combinations such as, Mathematical Analysis, Real Analysis, Complex Analysis, Numerical Analysis and Functional Analysis, to loosely indicate areas of mathematics that rely heavily on limiting processes and the real number system. This applies in particular to the foundations of the differential and integral calculus.

My first title led me to ponder the extent that mathematics had seeped into the popular imagination. Many who protest that they know no mathematics can read maps, and graphs, they can talk about speed, acceleration, inflation, betting odds, average temperatures, interest rates, and percentage increases, and when it suits they can follow logical arguments, while only vaguely associating these concepts with mathematics. But, for many, mathematical development ceases at the gates of academia. This is unfortunate as today there is an acute need for an educated public with a confident and non-trivial understanding of mathematics.

Mathematics has always played a role in civilised society and most are aware of its benign contributions. Unfortunately, over the last quarter century partially educated pseudo-experts, often in managerial positions, have realised that it is easy to associate numbers with different human activities and to derive conclusions with little supporting evidence. Ironically, these developments are possible because mathematicians played a prominent role in developing three areas which facilitated the numerical analysis and interpretation of large quantities of data: statistics, computer science and the internet. The conclusions of these present day alchemists are rarely challenged by a unquestioning public and, unfortunately, the consequences may be serious. This is not a modern phenomenon. Throughout history, knowledge and ignorance have both been employed as a means of control. Priests in ancient Egypt controlled the calendar because it gave them power and the greatest promotor of statistics for purposes of political control in the modern era was Josef Stalin. Today's pseudo-experts rely on the mathematical ignorance of the masses and mathematicians should be alarmed, rather than flattered, by the widespread belief that everything can be measured and thus reduced to a number.

That mathematicians occupied over the centuries influential positions in politics, the church, the law, medicine, business, engineering, science, and the military is a tribute to the benefit of an education in clear thinking and abstract reasoning, a benefit recognised thousands of years ago by Socrates, Plato and Aristotle. Mathematical ideas evolved over considerable periods of time and the student of mathematics often follows, perhaps subconsciously, the path followed by those who developed the subject and may well struggle with the same difficulties that initially surrounded the introduction of new ideas. These were some of the thoughts that motivated our introduction of historical material into this text, later we used the opportunity to provide a background for mathematics. Nevertheless, no matter how much we read and no matter how strong our imagination we can never fully appreciate the overall atmosphere of a bygone age, but we can glimpse how things were, and we can be inspired. In the following paragraph we give one brief example.

The ten years, 1789-1799, was a turbulent period in French history, At the beginning of the decade the French monarchy of Louis XVI was in place. This was followed by the French revolution and by the end of the decade Napoleon was the uncrowned emperor of France. The revolution, usually portrayed only as a brutal class war, aimed at fundamentally changing society. During this chaotic period three mathematical legends, Lagrange, Laplace end Legendre, were at the height of their powers and lived in Paris while the future legends Fourier and Poisson were students in the same city. These mathematicians subscribed to the revolutionary philosophy of building a modern egalitarian state using science and reasoning. They maintained and developed, in unbelievably uncertain circumstances, their mathematical interests and became involved at a highest level in the revolution. All survived, and during that decade the three established mathematicians served on the committee that established the metric system of weights and measures while publishing important mathematics. The National Convention enacted a bill on 30 October 1794, setting up the Ecole Normale to train a new type of teacher and each district in France was to send a small number of its most talented citizens to Paris to hear the experts lecture on their subject. Laplace, after diplomatically avoiding the call to be a talented citizen, was nominated as an expert and gave the inaugural lecture in mathematics to seven hundred mature students on 20 January 1795. His approach was revolutionary, he aimed at showing the most important discoveries, their principles, the circumstances that led to their birth, the most direct route to them and the procedures for making new discoveries. The ten lectures given by Laplace, were too advanced for his audience even though he used no equations in his earlier lectures and tried to explain abstract concepts in the language of his audience. They were published later and set the standard for public education in mathematics at second and third levels in France.

This book is devoted to the different aspects of one variable calculus. There are overlaps between the different chapters but each has its own dominant theme. The traditional differential calculus is developed in Chapters 10 and 11 and the integral calculus in Chapters 13 and 14. All other chapters are devoted to the background necessary to understand these chapters. Chapters 1 and 2 are an informal and intuitive introduction to mathematics, number systems are examined in Chapters 5 and 12, sets in Chapter 6, functions in Chapters 3 and 4, continuous functions in Chapter 9, and sequences and series in Chapters 7 and 8.

The features that distinguish this book from the many other books on the calculus are, hopefully, apparent to even casual readers as they proceed. We aim at a comprehensive, efficient, and rigorous treatment and introduce all the concepts that this necessitates. Functions, increasing (and decreasing) sequences, and bounds (upper and lower) play an important role in our study. Functions were often essential in allowing us to express ourselves clearly, the intuitive notion of increasing (and decreasing) convergent sequence helped us present a rigorous treatment of arbitrary sequences and series without resorting to the $\epsilon-\delta$ approach of Weierstrass. Increasing sequences of dyadic rationals allowed us to construct the real numbers without the use of either Dedekind sections or Cauchy sequences. By concentrating on polynomials, the exponential function, and power series, the fundamental concepts became more transparent. Moreover, students who persisted with our streamlined approach had little difficulty absorbing, and using in later courses, the traditional material omitted here.

There are diverse audiences for this book and, unlike the generic twentieth century encyclopaedic tomes on The Calculus, this offering is not a catchall that can be recommended ubiquitously. The serious study of mathematics takes time, as does the study of any discipline, this is a fact and it should be anticipated. A positive predisposition to mathematics and a persistent commitment to mastering the material are the main prerequisites required of the reader. Those who have left the world of structured mathematics courses behind but who now, as adults, wish to study and understand mathematics may be interested in this book. The level of abstraction varies considerably between chapters and the nature of the discipline suggests that the lone reader, while following the structure of a formal course, should choose the pace that they find suitable. I recommend that these readers skim over Chapters 5, 6 and 12 on a first, and perhaps a second reading, and dip back into the material in these chapters when they feel ready to do so. The 200+ exercises, most of which were developed and tested in tutorials or as homework, were chosen because they forced understanding and are an essential part of this book. Some are routine, but many are not, and although most relate to earlier material this is not always the case, and to help the reader appreciate the unexpected a few have been placed strategically in unorthodox locations. Mathematics should be read backwards and forward. Each attempt at an exercise should lead to a focused perusal of at least one chapter and result in the understanding of at least one fact that was read, or merely passed over, a dozen times previously. This is how we make progress in mathematics. We have provided solutions to some exercises, partial solutions to others, and no solutions to a number.

This book may be used as a text in a standard one variable calculus or analysis course. But we only recommend it as the main text when the teacher has first experienced it as a supplementary text, perhaps to a small group. It is not our intention to appear elitist with this comment. We are quite serious. A casual reading may create a false impression about the pace and difficulty of the material. In particular, it may well hide the degree of involvement required of the teacher and the level of participation demanded of the student, both of which are essential if the approach we advocate is to succeed. Students with an aptitude, perhaps latent, for mathematics, intelligent students who have convinced themselves that mathematics is not for them, and weaker students with a sympathetic and experienced guide, appear to benefit most immediately from this book. The modestly successful average student, who has learned to cope with rote learning, may require additional motivation.

9.3. Review by: P N Ruane.
Mathematical Association of America Reviews (22 July 2013).

This book is based on a first year course for students of Economics and Finance at University College of Dublin, and it was part of a programme of Business Studies that required 'sophisticated mathematics' during each year of undergraduate study.

Because mathematics courses for engineers, accountants, economists etc, are all too often of the handle-turning, 'ask-no-questions' variety, it's very refreshing to discover that Sean Dineen's philosophy is the complete antithesis of such educational short-sightedness. In fact, Dineen's approach is a true reflection of the attitude to mathematics shown by Laplace in his open lecture of 20 January 1795. For this book, it means that:

1. The vast majority of topics are set in an historical context.

2. Mathematics is seen as a way of knowing - not just as a body of knowledge.

3. Readers are led to explore the inter-connectedness of mathematical ideas.

4. The book will prepare students for ongoing mathematical study.

Basically, this book is suited to a course in one variable calculus and real analysis; but it would be wasteful to confine its use to students of business studies. For example, it is ideally suited for use on courses for future high school teachers, and it would lay good foundations for first year maths majors.

The first chapter centres upon an exploration of quadratic equations from the perspective of the formula that provides their roots. It examines the conditions under which this formula is applicable, and eventually considers analogous results for cubics. Because the approach is simultaneously heuristic and strongly historical, students will have begun to develop an expanded vision as to how, and why, mathematics is created.

This philosophy pervades the second chapter, which examines the role of diagrams and graphs in the mathematical thinking. Subsequent chapters then introduce the concepts of sets, functions and relations that are essential for the development of analytical ideas. Again, the presentation exemplifies a good combination of both horizontal and vertical thematic sequencing of ideas.

Real analysis begins (and continues) by avoiding the Weirerstass $\epsilon-\delta$ definitions. Convergence of real sequences is expressed with respect to lower and upper bounds, whilst continuity of a real valued function is defined in terms of its action upon a convergent sequence. Much of the material on differentiation and integration is standard, but the treatment is very much in the spirit of the aforementioned principles.

There's no way round the fact that real analysis ($\epsilon-\delta$ or not) remains a significant hurdle for incipient mathematicians - and this book is no easy option. Moreover, there are three challenging chapters on constructional approaches to number systems that (although they may be omitted on a first reading) reinforce the theoretical underpinnings of real analysis. For example, the chapter that defines real numbers in terms of convergent sequences of dyadic numbers would challenge any honours mathematics student.

One other attractive feature of this book is the nature of the exercise sets. The problems are truly instructional and a sufficient number of them are supplied with solutions. Overall, this book is highly recommended as a refreshingly different introduction to undergraduate mathematics.

9.4. Review by: Tom Carroll.
Irish Mathematical Society Bulletin 69 (2012), 57-59.

The book under review is a first course not only in analysis and calculus but in the culture of mathematics. It grew from lecture notes for a mathematics course aimed at Economics and Finance students at University College Dublin and caters not only for mathematics students but for students whose area of primary interest lies outside mathematics. It is clear that the author believes that all students, including those who would typically be classed by a mathematics department as taking a 'service course', should be expected to understand the principles of mathematics and be skilled in their use.

Dineen paints on a broad canvas. The topics standard to all calculus and analysis textbooks are covered completely and in detail - number, function, limits and continuity, sequences and series, differentiation, integration, applications - with the approach being thorough right from the beginning. The reader is encouraged to think about each topic from different points of view. Rather than assuming the role of the omniscient author who providentially introduces material in advance of needing it, Dineen deals with issues only as they arise and introduces new mathematics only as it is needed. For example, though proofs are centre stage throughout the book, readers pick up proof techniques by degrees and in a manner commensurate with their growing mathematical maturity and confidence. That this approach works is partly due to the historical narrative which runs parallel to the mathematics and which explains how central concepts, such as number, function, limit, came into focus only gradually and were used implicitly long before their modern definitions solidified, often long before it was even realised that a precise definition might be needed. Pen pictures of the main contributors to the development of mathematics give the narrative a human feel and reinforce the message that understanding this mathematics takes time and commitment. Teachers' expectation of engagement on the part of students needs to be matched by a reciprocal commitment to engage with students and to bring students along with them. Dineen does this through a non-linear mathematical narrative. In the first chapter on Quadratic Functions, the rules of algebra, cancelation, square roots, functions are used freely, even if they will be formally introduced only later on. Though the exponential function and the logarithm function (modulo the intermediate value theorem which is proved later) are first formally defined in Chapter 7, these functions are introduced informally in Chapter 2 and are used freely throughout the book. Real Numbers feature right from the start - that positive numbers have a square root is used in Chapter 1, the completeness axiom is the basis of Chapter 6 where all the main properties of supremum and infimum of sets of real numbers are proved – even if the completeness axiom and the construction of the real numbers comes later in Chapter 12. It is in this sense that the narrative is 'non-linear'. It has taken great care and thought on the part of the author to ensure that this approach works logically, which it most certainly does. Dineen is frank about his approach. He introduces concepts gradually, informally at first, with an emphasis on understanding rather than absolute rigour. He aims to 'blend intuitive techniques and rigorous definitions' with the rigor coming later, often motivated by the historical realisation that clarity and precision would be essential if further progress was to be made (c.f., for example, Berkeley's criticisms of Newton and Leibniz's calculus, which Dineen discusses in detail). Dineen's approach steers a careful course between the ubiquitous calculus tome and a potentially dry first course in mathematical analysis and, in so doing, is more reflective of the way we learn mathematics. Let me be entirely clear that, though the mathematical development often moves ahead of itself only to regroup later, the narrative is entirely consistent and leaves no loose ends.

To conclude, a technical word or two about the author's mathematical choices. The rational numbers are constructed from the positive rational numbers which are in turn constructed directly from the natural numbers, so that the integers come after the rationals in Dineen's development. This works very well and is, of course, a perfect opportunity to bring in equivalence relations. Countability is covered in the context of number and function. Analysis is based on sequences, which avoids any $\epsilon-\delta$ arguments. In fact, Dineen avoids $\epsilon$ entirely by restricting himself first to monotonic sequences, defining the limit of a bounded monotonic sequence to be the supremum if it is increasing, or the infimum if it is decreasing, of the set of real numbers which occur in the sequence. Having consolidated this notion through a variety of examples and results, Dineen defines a general sequence to be convergent if it lies between an increasing and a decreasing sequence which are convergent and have the same limit. All aspects of infinite series are covered in detail. From here he naturally defines continuity of a function at a point to be sequential continuity. I particularly enjoyed Chapter 10 on the construction of the real numbers using sections of the dyadic rationals and a 'bisection principle'. The last sections of the book cover first the derivative and its applications, then the Riemann integral for continuous functions and its applications. A notable feature of the book are the numerous well thought out, interesting exercises at the end of each chapter, with solutions provided at the end of the book.

10.1. From the Publisher.

The use of the Black-Scholes model and formula is pervasive in financial markets. There are very few undergraduate textbooks available on the subject and, until now, almost none written by mathematicians. Based on a course given by the author, the goal of this book is to introduce advanced undergraduates and beginning graduate students studying the mathematics of finance to the Black-Scholes formula. The author uses a first-principles approach, developing only the minimum background necessary to justify mathematical concepts and placing mathematical developments in context.

The book skilfully draws the reader toward the art of thinking mathematically and then proceeds to lay the foundations in analysis and probability theory underlying modern financial mathematics. It rigorously reveals the mathematical secrets of topics such as abstract measure theory, conditional expectations, martingales, Wiener processes, the Itô calculus, and other ingredients of the Black-Scholes formula. In explaining these topics, the author uses examples drawn from the universe of finance. The book also contains many exercises, some included to clarify simple points of exposition, others to introduce new ideas and techniques, and a few containing relatively deep mathematical results.

The second edition contains numerous revisions and additional material designed to enhance the book's usability as a classroom text. These changes include insights gleaned by the author after teaching from the text, as well as comments and suggestions made by others who used the book. Whereas the revised edition maintains the original approach, format, and list of topics, most chapters are modified to some extent; in addition, the rearrangement of material resulted in a new chapter (Chapter 9).

With the modest prerequisite of a first course in calculus, the book is suitable for undergraduates and graduate students in mathematics, finance, and economics and can be read, using appropriate selections, at a number of levels.

10.2. From the Preface.

Comments from different sources, experience in using the first edition as a class text, and the opportunity to teach a preliminary course in analysis to students who would subsequently use this book all contributed to the changes in this second edition. The analysis experience resulted in Analysis: A Gateway to Understanding Mathematics published by World Scientific (Singapore) in 2012. While maintaining the original approach, format, and list of topics, I have revised to some extent most chapters. I found it convenient to rearrange some of the material and as a result to include an additional chapter (Chapter 9). This new chapter contains material from Chapters 6 and 7 in the first edition and, additionally, a construction of Lebesgue measure using dyadic rationals and a countable product of probability spaces.

A brief paraphrasing of essentially one paragraph of the original preface is included here to help the reader navigate the second edition. Students of financial mathematics may wish to follow, as our students did, Chapters 1-5; Sections 6.1, 6.2, 6.3 and 7.4; the statements of the main results in Sections 6.3, 6.4, and 7.5; and Chapters 8 and 10-11. Students of mathematics and statistics interested in probability theory could follow Chapters 3-7, Section 8.2 and Chapters 9 and 10. Students of mathematics could follow Chapters 3-6 and 9 as an introduction to measure theory. Chapter 12 is, modulo a modest background in probability theory, a self-contained introduction to stochastic integration and the ltô integral. Finally anyone beginning their university studies in mathematics or merely interested in modem mathematics, from a philosophical or aesthetic point of view, will find Chapters 1-5 accessible, challenging and rewarding.

11.1. From the Publisher.

Multivariate calculus can be understood best by combining geometric insight, intuitive arguments, detailed explanations and mathematical reasoning. This textbook has successfully followed this programme. It additionally provides a solid description of the basic concepts, via familiar examples, which are then tested in technically demanding situations.

In this new edition the introductory chapter and two of the chapters on the geometry of surfaces have been revised. Some exercises have been replaced and others provided with expanded solutions.

Familiarity with partial derivatives and a course in linear algebra are essential prerequisites for readers of this book. Multivariate Calculus and Geometry is aimed primarily at higher level undergraduates in the mathematical sciences. The inclusion of many practical examples involving problems of several variables will appeal to mathematics, science and engineering students.

11.2. Review by: Peter Shiu.
The Mathematical Gazette 100 (547) (2016), 181-182.

Mathematical ideas often come from intuition and insight arising from the desire to solve physical problems. If an idea works then we develop it further for implementation; at this stage, the imposition of a logical argument or a precise language only hinders progress. The question of why such a method works will eventually be asked, and a proper understanding arising from proofs based on fundamental principles often enhances the method so that it becomes a powerful tool capable of tackling a variety of problems. This is perhaps particularly so for the calculus, which was widely and successfully used as a method to solve physical problems before the proper understanding of the notion of a limit. A university course in mathematics involves the study of analysis, as well as 'advanced calculus' the former gives the theory on limiting processes while the latter deals with the generalisation and manipulative techniques in the calculus of several variables. There is no shortage of texts catering for students in both these areas.

The book being reviewed is unusual in that it provides an in-depth study of advanced calculus. It is pitched at a level which is perhaps too tough for many undergraduates, but the book is very useful for those who wish to learn the theory properly. The necessary notation and terminology are set out in an initial chapter. and the remaining 17 chapters cover three main areas: differential calculus on open sets and surfaces; integration theory; geometry of curves and surfaces.

Level sets. tangent spaces, maxima and minima are considered in the differential calculus, with frequent reminders to the reader on the relevance of a set being open or being compact. Integration theory covers line and multiple integrals, with emphasis on the concepts of parametrisation and oriented surfaces; the theorems of Green and Stokes and the divergence theorem of Gauss are presented in the context of the fundamental theorem of calculus in higher dimensions. The part on the geometry of curves and surfaces is a useful introduction to differential geometry. The topics include curvature. torsion. geodesics. and the derivation of the Frenet-Serrat equations; the Gauss-Bonnet formula relating the Gaussian curvature to the Euler characteristic of a surface is also included.

The sophisticated presentation of the subject matter means that this book is not suitable for someone who just wants to acquire methods and techniques in advanced calculus. However, the book is very clearly written-the theory is nicely presented with important topics being well explained and illustrated with examples. It is particularly suitable for someone who is familiar with the basics of the material and wants to study the theory. Each chapter begins with an outline of its content, and ends with suitably constructed exercises, with solutions given at the end of the book. Moreover, the introduction of a new concept is followed by simple or familiar worked examples to accompany the theory, and the text contains many diagrams to illustrate the problems being discussed. Besides being a useful book for the keen student, it is also an excellent reference text on multivariate calculus and the basics in differential geometry.