# Elon Lima's Books

1. Topologia dos Espaços Métricos (1958), by Elon Lages Lima.
1.1. Review of the 1954 mimeographed notes: by Jacques Riguet.
Mathematical Reviews MR0077109 (17,991b).

First part of the general topology course given by the author as part of his teaching of higher geometry during the year 1954 for 2nd and 3rd year mathematics students at the Faculty of Philosophy of Rio de Janeiro. This presentation, clear and precise, interspersed with examples and exercises, follows Nicolas Bourbaki fairly closely and can constitute an excellent introduction to the latter's treatise on general topology. This is, moreover, a trend that seems to be taking shape in various countries. This appeal to the good services of metric spaces, places where the appeals of the student to his Euclidean intuition will still be heard, after having been used by him as preliminary stages towards the rougher paths of general topology. The list of the various chapters which constitute this course is given as follows: 0) Preliminaries. 1) Metric spaces. 2) Continuous functions. 3) Spheres and open sets. 4) Closed. 5) Sequels. 6) Convex sets and topology. 7) Subspaces. 8) Complete space. 9) Separable space. 10) Compacts. 11) Homeomorphisms. Appendix: Recommended reading.
2. Teoria Global dos Mínimos e Máximos de uma Função Real (1959), by Elon Lages Lima.
2.1. Note.

No details found.
3. Introdução às variedades Diferenciáveis (1960), by Elon Lages Lima.
3.1. Review by: William S Massey.
Mathematical Reviews MR0155332 (27 #5266).

This is an introductory text based on a course of lectures given by the author in 1959. The prerequisites, in addition to a knowledge of advanced calculus, are a small amount of linear algebra and general topology. Tensor analysis is not used.

There are three chapters. The first is concerned with differentiable functions whose range and domain are open subsets of Euclidean space, and culminates in a proof of the inverse function theorem. The second chapter is concerned with "regular surfaces in Euclidean space", i.e., differentiable manifolds imbedded in Euclidean space. The main theorem proved in this chapter is the existence of a tubular neighbourhood for such an imbedded manifold which is a fibre bundle over the manifold. The last chapter introduces differentiable manifolds in the usual abstract sense (without any imbedding in Euclidean space, as in Chapter 2). The principal result of this chapter is a proof of the Whitney imbedding theorem for compact manifolds of class two.

The value of this booklet is considerably enhanced by many illustrative examples which are discussed in detail and numerous exercises for the reader. There is also an informal discussion of some of the basic problems of differential topology and mention of the work of Milnor, Smale, and Kervaire.

This book will help considerably to smooth the pathway for Portuguese-speaking students who want to learn modern mathematics.
4. Introdução à Topologia Diferencial (1961), by Elon Lages Lima.
4.1. From the Preface.

There is, in Mathematics, a very extensive and general discipline called Topology, which deals with the notion of continuous function in its broadest sense. That is, its scope is the category of all topological spaces. Now, despite the importance and suitability of this concept, it is impossible to demonstrate a non-trivial theorem on any topological space, given the extreme variety of species contained in the genus in question. Thus, to be able to work honestly, the topologist needs to impose restrictions on the type of spaces he will consider. According to the nature of these restrictions, and also according to the type of auxiliary tools used, we can highlight the following branches of Topology.

a) General Topology. Here, the restrictions imposed on topological spaces consist of adding one or more axioms that refer directly to the open sets of the space, either guaranteeing the existence of a sufficiently large number of them (separation axioms), or preventing that there are too many open sets (compactness), etc. Anyway, what characterises General Topology from a global point of view, the basic notion in the topology of polyhedra is the concept of incidence (is this segment the side of which triangles?, is this triangle the face of which tetrahedrons?), through which one passes from a polyhedron to its abstract scheme, formed by a finite number of objects, linked by a unique relationship, of incidence. The study of this scheme is, then, a combinatorial problem, hence the name of this branch of Topology.

c) Algebraic Topology. What distinguishes Algebraic Topology from General Topology and Combinatorial Topology (as well as from Differential Topology, which we will discuss below), is not the nature of the spaces it considers, nor the additional structures that exist or do not exist in these spaces, but first of all, the working method. Algebraic Topology is sometimes more general than General Topology because it is possible to demonstrate non-trivial theorems that apply to all topological spaces. On the other hand (and here lies the reason for the above paradox), Algebraic Topology is not exactly a branch of Topology, but a transition method from it to Algebra. The basic idea is as follows. Algebraic structures are, in general, simpler than topological structures. For example, determining whether two given abelian groups are isomorphic or not (especially when they are finitely generated) is almost always a simpler problem than determining whether two given topological spaces are homeomorphic or not. Once this maxim is admitted, any process that allows systematically replacing topological spaces by groups (for example) and continuous functions by homomorphisms acquires great interest, so that if two spaces are homeomorphic, then the groups associated with them by this process are isomorphs. (One cannot expect the converse to be valid, as this would imply that the general structure of groups would be as complicated as the general structure of topological spaces). Such a process is what is called a "functor" defined in a topological category, with values in an algebraic category. Algebraic Topology deals with the study of these functors whose best-known examples are the homology and homotopy groups.

d) Differential Topology. This branch of Topology is characterised by the additional structure of the topological spaces it deals with (differentiable manifolds) and, consequently, by the working methods, which are, initially, those of the classic Differential and Integral Calculus, assisted by the results of Algebraic Topology. This is a return to the origins of the subject because, as is known, in the first of the memoirs in which Poincaré developed the bases of modern Topology, the topological spaces he considered were submanifolds of Euclidean space, "homeomorphism" for him was what today we call "diffeomorphism", and the spirit in which Poincaré wrote this memoir, undoubtedly motivated by problems of Analysis, is the forerunner of the current spirit of Differential Topology. It was later that, due to purely technical reasons, he abandoned the differential point of view, in favour of the combinatorial method, for lack of adequate resources of General Topology, among which the "passage from the local to the global" allowed by the recent notion paracompact space.

These notes represent an elaborated version of a course given at the 3rd Brazilian Colloquium of Mathematics, in Fortaleza, Ceará. They are what the title says: an introduction to Differential Topology. They are not entirely self-contained, since they assume that the reader is familiar with the most elementary notions about Differentiable Manifolds, which can be found in the bibliography cited in Chapter I.

Chapter II contains the homotopic classification of continuous maps of a manifold of dimension $n$ on the sphere $S^{n}$, Chapter III demonstrates that the sum of the singularity indices of a "generic" vector field over a compact manifold $M$ is an invariant of $M$, called the Euler characteristic. Chapter IV demonstrates the Integral Curvature Theorem. To arrive at these results, several important ideas and techniques are introduced, such as the approximation of continuous functions and homotopies by differentiable ones, the notion of degree, the integration of differential forms, the concept of transversality, the intersection number of two submanifolds, etc.

The implacable rigidity of the deadline in which these notes had to be ready did not allow them to be presented, as was the initial intention, at a more leisurely pace. In this particular, the most serious sin is the lack of adequate development for the 3 examples at the end of Chapter III.

Brasilia, June 21, 1961
Elon Lages Lima

4.2. Review by: A M Rodrigues.
Mathematical Reviews MR0159341 (28 #2558).

An introduction to differential topology with emphasis on results and proofs of a geometric nature; algebraic topology is avoided as much as possible.

Chapter I introduces several basic results on differentiable manifolds without proof, for later reference. Chapter II deals with the notion of the degree of a mapping of a differentiable manifold into another of the same dimension. The definition is given first for differentiable mappings and then generalised to the continuous case. The chapter ends with the classification of the homotopy classes of continuous mappings of a compact differentiable manifold $M^{n}$ into the sphere $S^{n}$ by means of the degree. The next chapter is concerned with singularities of vector fields and with the proof that the sum of the indices is independent of the vector field when the manifold is compact and the singularities are isolated. In the fourth and last chapter the theorem of the Curvatura Integra of Hopf is proved. The volume is carefully written and the exposition is very neat throughout; many examples are treated in detail.
5. Cálculo Tensorial (1965), by Elon Lages Lima.
5.1. From the Preface.

With the exception of the final chapter, these are lecture notes from a course I taught twice, in 1960 and 1962. The opening three chapters are based on an essay from the first course by J Ubyrajara Alves. The fourth chapter, given only the second time, was written by Alciléa Augusto. J B Pitombeira collaborated in writing the first chapter. These friends, whom I now thank, are responsible for the appearance of the notes, but certainly not for the errors and defects they contain, of which I am the sole author.

The idea here is to present an introduction, modern but without "modernisms", to Calculus and Tensor Analysis. With regard to the first (where we restrict ourselves to real vector spaces of finite dimension and thus take advantage of the various simplifications that these hypotheses entail) the exposition is enough for the purposes of Differential Geometry and Analysis. On purpose, the tensor products of modules were not mentioned, and thus the introduction presented here serves only as a motivation for the general theories of Homological Algebra. As for Tensor Analysis, the surface has barely been scratched. An introductory beginning was made, in Chapter 4, leading to Stokes' Theorem as a final result and, in Chapter 5, the Frobenius Theorem was demonstrated from the point of view of the Lie bracket of two vector fields. The groundwork was solidly laid, with discussions of fundamental concepts such as differentiable manifolds, orientability, partitions of unity, integration of differential forms, exterior differential, and Stokes' Theorem. But, in keeping with the essentially introductory character of the notes, no deeper development is attempted. At the end of the work, a list of bibliographic indications is presented, as a relief of conscience.

Classical Tensorial Calculus, in its strictly algebraic part, is little more than a repertoire of trivia. This is reflected in the nature of Chapter 2, where an intrinsic and conceptual presentation of tensors is given. There is a conspicuous absence of "cunning" theorems there. It is interesting to contrast this fact with Chapter 3, where Grassmann Algebra is studied and where interesting applications to the Theory of Determinants and Geometry arise. In Chapter 4, which relates to Differential Geometry the algebraic foundations laid in the first three chapters, the only tensors used are the antisymmetric ones from Chapter 3.

In these notes, the famous "Kronecker delta" never appears, nor is the so-called "Einstein convention" adopted. The first could bring some small advantage by saving us from writing two or three more words. The second is not only unnecessary but downright absurd. The fuss it causes isn't worth the trouble of writing a ∑ here and there. By setting these notations aside, we pay our simple homage to Kronecker and Einstein, who must be remembered for more serious reasons. On the other hand, we adopted the classic use of, whenever possible, placing repeated indices at different heights. By doing so, mistakes are avoided when writing the sums: first put the main letters and then fill in the indices, so that "it works out".

Brasilia, November 6, 1964
Elon Lages Lima

5.2. Review by: Maurício Matos Peixoto.
Mathematical Reviews MR0196644 (33 #4831).

This is a carefully written, well motivated introduction to tensors, highly recommended for beginners, "modern without modernisms'' (as the author puts it). The work is divided into five chapters: Vector spaces, Multilinear algebra, Exterior algebra, Differential forms (including the theorem of Stokes) and Differential systems. The last chapter, which goes beyond the usual presentation of tensors, treats such topics as Lie brackets, commuting vector fields, and the theorem of Frobenius about complete integrability.
6. Análise Geométrica (1969), by Elon Lages Lima.
6.1. Review by: Editors.
Mathematical Reviews MR0393373 (52 #14183).

This is an account of the theory of functions of several real variables in intrinsic (coordinate-free form). The author presupposes only some linear algebra and elementary topology of Euclidean spaces but he points out that some of the theorems and proofs carry over to the calculus in Banach spaces.

Chapter headings: (1) Differentiable mappings; (2) Examples; (3) The differentiability class $C^{k}$; (4) The chain rule; (5) The mean value inequality; (6) Integrals; (7) Partial derivatives; (8) A theorem of Schwarz; (9) Taylor's formula; (10) Implicit functions; (11) Change of variable in multiple integrals.
7. Elementos da Topologia Geral (1969), by Elon Lages Lima.
https://loja.sbm.org.br/elementos-de-topologia-geral.html
7.1. Description of the 2014 edition by the Brazilian Mathematical Society.

To take full advantage of this book, you must have had an initial contact with the language, the basic concepts and the elementary properties related to sets and functions. The notion of a real number is also required, especially the concept of the supremum, or upper bound, and the lowest, or lower bound, of a limited set of real numbers. These are the two pieces of advice from Elon Lages Lima, who reminds readers that the work expresses his own ideas about general topology. Topics related to these prerequisites appear in the first part of the book.

To facilitate the contact with the theme, Elon Lima gives the theorem demonstrations and the discussions of examples in detail; he illustrated the basic concepts and fundamental results with examples and counterexamples; and restricted generality and abstraction. This means that uniform filters and structures appear in the exercises, separation axioms are discussed only when they need to be used, and metric spaces are often considered and examined in private study.
8. Análise no Espaço $\mathbb{R}^{n}$ (1970), by Elon Lages Lima.
8.1. Description of the 2013 edition by the Publisher.

This book, which deals with differential and integral calculus for functions of several variables, can be used as a continuation of the book Real Analysis Vol.1, written by the same author. This book contains a brief and rigorous presentation of the most important facts about to differentiability in $n$-dimensional space.
9. Logarítmos (1973), by Elon Lages Lima.
9.1. Description of the 2019 edition by the Brazilian Mathematical Society.
https://loja.sbm.org.br/logaritmos.html

Before calculating machines and computers, logarithms were the tools that simplified arithmetic calculation because they allowed one to perform, quickly and accurately, complicated operations such as the multiplication of two numbers with many digits, or a potentiation with a fractional exponent. In the book "Logarithms," Elon Lages Lima deals in an elementary way with the applications of logarithms in mathematics teaching, such as in the calculation of interest or continuous losses, for example when investing capital $c$ in a business and there is a loss or gain of $k$% a year, considering, above all, that there are good and bad deals.

In a clear and accessible way for readers, Professor Elon introduces the traditional definition of logarithms and the concept of logarithmic functions. This book can be used in high school and graduation, especially, because it suggests topics that can be addressed in the classroom. In addition, it brings tables that can be used to illustrate how numerical calculations were made with logarithms and exponential functions.
10. Áreas e Volumes (1973), by Elon Lages Lima.
10.1. Note.

No information found.
11. Variedades Diferenciáveis (1973), by Elon Lages Lima.
11.1. Review by: Robert E Stong.
Mathematical Reviews MR0526597 (80i:57001).

This book is an introductory text about differentiable manifolds, quite reminiscent of the first chapter of J Milnor's unpublished lecture notes, "Differential Topology'', of 1958.

The author begins with the implicit function theorem of calculus and using it obtains Thom's transversality theorem and the immersion and imbedding theorems of Whitney. The presentation is leisurely, with many pictures and examples. Along the way, one picks up sidelights about Lie groups, vector fields, Riemannian metrics, and the topology of function spaces.
12. Álgebra Exterior (1973), by Elon Lages Lima.
12.1. Description by Publisher of the 2017 edition.

Exterior Algebra  is, from the purely algebraic point of view, the study of alternating multilinear applications and their ramifications. From the geometric point of view, it deals with the $p$-dimensional vectors, which were originally conceived by H Grassmann. The presentation contained in this book is elementary, being accessible to readers who are familiar with the basics of Linear Algebra.

12.2. Description of the 2017 edition by the Brazilian Mathematical Society.
https://loja.sbm.org.br/algebra-exterior.html

What is the application of exterior algebra? This field of study of alternating multilinear maps and their ramifications has utility for differential geometry, algebraic topology, mechanics, and partial differential equations. Its knowledge is essential for anyone interested in mathematics.

The introduction made by Elon Lages Lima is focused on finite-dimensional vector spaces over the real field. It seeks to present the basic concepts and results of the theory in a simple way. His approach involves topics such as permutations and sequences, multilinear maps, determinants, outer and inner product of $r$-vectors, outer forms, induced maps and the Grassmann algebra.
13. Espaços Métricos (1975), by Elon Lages Lima.
13.1. Review by: Editors.
Mathematical Reviews MR0654506 (83d:54001).

This book was written to serve as a textbook for a course on metric spaces, as an introduction to topology. Table of contents: 1. Metric spaces; 2. Continuous functions; 3. Basic notions of topology; 4. Convex sets; 5. Limits; 6. Uniform continuity; 7. Complete metric spaces; 8. Compact metric spaces; 9. Separable spaces.

13.2. Description of the 2020 edition by the Brazilian Mathematical Society.
https://loja.sbm.org.br/espacos-metricos.html

This book, winner of the Jabuti Prize, is essential for studies in the final years of undergraduate mathematics. Students with notions of analysis will better understand the applications of the theory of metric spaces. Elon Lima introduces the basic language of topology, connected sets, limits, and complete and compact metric spaces.

Applications such as the fundamental theorem of algebra, the existence of continuous functions with no derivative at any point, the Peano curve, Picard's theorem on existence and uniqueness of solution for ordinary differential equations, Montel's theorem related to families are described, normal analytic functions, the Stone-Weierstrass theorem and the Hilbert cube as a universal separable space. These examples seek to show the strength and multifunctionality of the theory.
14. Curso de Análise Vol 1 (1976), by Elon Lages Lima.
14.1. From the Publisher.

The author presents the language of sets and functions through a precise conceptualisation and logical systematisation of ideas. Its objective is to study the sets of real numbers and real functions of a variable. The given concepts are illustrated by examples and accompanied by many exercises of varying degrees of difficulty. The best use of this analysis course requires prior knowledge of calculus.

In addition to exposing general concepts and basic facts about sets and functions, real numbers, the foundations of their theory, sequences and series are presented; the topology of the line; the limits of functions; derivatives; the Riemann integral; and sequences and series of functions. The author adopted an informal and descriptive style in the first chapters and an axiomatic point of view in the others.

Contents of Vol. 1: I. Sets and functions; II. Finite, countable and uncountable sets; III. Real numbers; IV. Sequences and series of real numbers; V. Topology of the line; VI. Limits of functions; VII. continuous functions; VIII. Derivatives; IX. Riemann integral; X. Sequences and series of functions.
15. Grupo Fundamental e Espaço de Recobrimento (1977), by Elon Lages Lima.
15.1 Note.

16. Fundamental groups and covering spaces (2003), by Elon Lages Lima.
16.1. From the Preface:

The origin of this book is a set of notes for a course that I taught, years ago, in the Brazilian Mathematics Colloquium and, several times after that, to beginning graduate students at IMPA, Rio de Janeiro. Later on, the notes were revised and appeared as a book in the collection 'Projeto Euclides', published by IMPA. Since then the book has been used as an introduction to algebraic topology at many Brazilian universities and in other Latin American countries.

The subjects discussed here, fundamental group and covering spaces, are well suited as an introduction to algebraic topology for their elementary character, for exhibiting in a clear way the use of algebraic invariants in topological problems and also because of the immediate applications to other areas of mathematics such as real analysis, complex variables, differential geometry and so on.

This is an introductory book, with no claims of becoming a reference work. The appeals to facts of analysis and algebra that are made in the text are very few and their aim is to show connections with other disciplines. If the reader so wishes, these appeals may be skipped without harm to the understanding of the text.

It is a pleasure to extend my warmest thanks to Jonas Gomes, a very dear friend and colleague, who suggested the translation of the book into English and, to my great surprise and contentment, undertook the job himself with his habitual competence, recommending a few changes and additions, which I made with satisfaction.

16.2. Review by: Philip Maynard.
The Mathematical Gazette 88 (512) (2004), 359-360.

Here is an excellent self-contained introduction to fundamental groups and covering spaces. It assumes only a few basic notions from topology: topological space, Hausdorff space and compactness, for example. It is an English translation of a book originally appearing in the collection 'Projecto Euclides' published by IMPA. As such, the book has been used widely in universities of Latin American countries as an introduction to algebraic topology. The translation, by a friend of the author, is clear and concise, making the book very readable. Apart from its intrinsic beauty the subject matter of the book has applications in complex analysis, differential geometry, group theory and physics. Various examples of this fact are given throughout the book. At the end of each chapter there are a total of 144 questions (without answers). However, no results from the exercises are assumed in the main text. The book is based on a course given in the Brazilian Mathematics Colloquium and, several times after that, to beginning graduate students at IMPA, Rio de Janeiro. The book is in two parts suggested by the title: Part I contains five chapters on fundamental groups and part II has three chapters on covering spaces. ... Though the book has a definite textbook feel to it, it provides an absorbing read for anyone interested in an introduction to algebraic topology. The text is interspersed with over 70 illustrative examples and over 50 well-drawn explanatory pictures. Overall, the book is clearly written and has a good flow to it and succeeds admirably in the task it set itself.
17. Curso de Análise Vol 2 (1981), by Elon Lages Lima.
17.1. From the Publisher.

The focus of this second volume is the functions of several real variables. Initially, items of the topology of Euclidean space, real functions of $n$ variables and curvilinear integrals are discussed. Next, differential maps, multiple integrals, and surface integrals. To better apprehend and master these subjects it is necessary to be familiar with functions of one variable and notions of linear algebra.

The study of topology is done in more depth in later chapters. In them, questions appear that led Gauss, Riemann and Kronecker to address analysis problems with methods that became part of differential topology. The author observes that the infinite variety of topological types of subsets of the space $\mathbb{R}^{n}$, which will serve as domains for the functions studied, gives the book a more geometric content. While in the first volume the character was more arithmetic.

17.2. From the publisher of the 2016 edition.

Elon Lages Lima was Researcher Emeritus of the Instituto de Matemática Pura e Aplicada (IMPA) and a full member of the Brazilian Academy of Sciences and of the TWAS (Academy of Sciences for the Developing World). He was also Doctor Honoris Causa by the Federal Universities of Amazonas and of Alagoas and by the Universidad Nacional de Ingeniería del Perú, Professor Honoris Causa of the Federal Universities of Ceará and of Bahia, of the Universidade Estadual de Campinas, of the Pontificia Universidad Católica del Perú and the University of Brasília. Received the Order of Scientific Merit in the Class Grã-Cruz, from the Presidency of the Republic and the Award Anísio Teixeira, from the Ministry of Education and Culture.

17.3. Description of the 2012 edition by the Brazilian Mathematical Society.
https://loja.sbm.org.br/analise-real-volume-2-funcoes-de-n-variaveis.html

After presenting the notions of a variable in the first volume of this series, Elon Lima addresses, in this second publication, the differential and integral calculus of functions of $n$ variables. It seeks to show the differential calculus of maps of $\mathbb{R}^{m}$, or a subset, in $\mathbb{R}^{n}$ and of multiple integrals. For this, it begins with the study of continuous functions, contemplated in the first chapter on topology of Euclidean space. Next, topics related to paths in $\mathbb{R}^{n}$, real and implicit functions are structured.

The second part deals with the derivative as a linear transformation, the inverse application theorem, differential surfaces, multiple integrals and changes of variables. In addition to allowing access to these themes of real analysis, the book has 170 exercises whose complete solutions can be consulted in the last chapter.
18. Meu Professor de Matemática e Outras Histórias (1987), by Elon Lages Lima.
18.1. Description of the 2012 edition by the Brazilian Mathematical Society.
https://loja.sbm.org.br/meu-professor-de-matematica-e-outras-historias.html

It is necessary to inquire and question before making statements. And in this book, made up of small essays on elementary mathematics, Elon Lages Lima exposes simple questions, such as the meaning of equality; whether zero is natural or not; the notion of paradox; negative numbers; why $(-1)(-1) = 1$, even the most elaborate ones like the definitions of the numbers $\pi$ and $e$; the number of faces of a polyhedron and repeating decimals. "My Mathematics teacher and other stories" is essential for both math teachers and primary school or university students. Some topics can be used in group studies or seminars by university students who have an interest in teaching.

In addition to focusing on mathematical aspects, the author rescues the history of the third degree equation in a very interesting way, emphasising the most general equation in algebra and some graphs with simple, double and complex real roots. It situates the evolution of some mathematical ideas, Euler's Theorem about polyhedra, which provides a relationship between the number of vertices, edges and faces of a polyhedron, Pick's formula, which gives us a formula for calculating the area of polygons defined by points on a network, proportional quantities, and some concepts and controversies. Elon also comments on three problems addressed in the popular book "O Homem que Calculava," by Malba Tahan (pseudonym of Professor Júlio César de Mello), published in Brazil in the late 1930s and which aroused the interest of several generations in mathematics.
19. Análise Real (1989), by Elon Lages Lima.

Finite and infinite sets. Real numbers. Sequences of real numbers. Series of numbers. Some notions of topology. Function limits. continuous functions. Derivatives. Taylor's formula and applications of the derivative. The Riemann integral. Calculation with integrals. Sequences and series of functions.
20. Coordenadas no Plano (1991), by Elon Lages Lima.
20.1. Preface of the 2002 edition

As its title indicates, this book is an introduction to the study of geometry through the systematic use of coordinates. This method, developed by Pierre Fermat and René Descartes in the 17th century, establishes an equivalence between geometric statements and propositions related to numerical equations or inequalities. Assuming that the simplest and most basic facts of Plane Geometry are known, we will show how to interpret these facts algebraically, in the form of relations between coordinates of points in the plane.

In the First Part of the book, these relationships are established in a direct and elementary way, without resorting to additional concepts. In the Second Part, the notion of vector is introduced, through which Algebra, in addition to being an auxiliary instrument of Geometry, becomes part of it, allowing operations with geometric entities to be carried out. The basic ideas and techniques of Vector Calculus are developed, with some applications to Geometry. In the Third Part, coordinates and vectors are used for an elementary and efficient treatment of the theory of geometric transformations, a topic of the highest relevance in the modern study of geometry.

In this new edition, we have included the solutions to the proposed problems, considering that reading rigorously written solutions is important for teacher training. This part was part of the book "Problemas e Soluções", published in 1992 with the support of Fundação Vitae.

It is not too much to insist that, before looking at the solution, the reader must make a serious attempt to arrive at it independently. If you can, compare your solution with the one presented here; there are almost always several paths to the same place. But, even if it does not have the desired success, the effort made brings several beneficial results: it serves to rethink and fix some concepts, to isolate the difficulties, it can serve to obtain partial answers (or victories) and it certainly helps to better understand the solution to the problem book.

There are no figures in the exercise statements, as drawing them is part of the required work. Some of these figures appear here along with the solutions. Others don't, but that doesn't mean they aren't necessary. It is still up to the reader to make the figures referring to the exercises, executing only those that have a strictly algebraic character.

Once again, I record here special thanks to my colleague Paulo Cezar P Carvalho for his valuable collaboration.

Rio de Janeiro, 4 October 2002
Elon Lages Lima
21. Medida e Forma em Geometria (1991), by Elon Lages Lima.
21.1 From the Preface.

This booklet is a re-edition, modified and greatly enlarged, of another one I wrote 20 years ago and which was published, in successive printings, by the Brazilian Mathematical Society.

The reformulation consists of the addition of a new chapter, several historical notes, new exercises and a general revision of the text. It was made with a view to the training course for teachers of secondary schools, the first phase of which took place in January 1991, under the sponsorship of VITA.

Description of the 2011 edition by the Brazilian Mathematical Society
https://loja.sbm.org.br/medida-e-forma-em-geometria.html

Who would have thought that the measurement of lands could have any connection with geometry? But it does. The relationship between both is explained by the historian Herodotus, in the fifth century (BC), who attributed the origin of geometry to the Egyptians. He says that landowners in that period paid a tax proportional to the area of each lot they owned. However, the floods of the Nile River made part of the farmers' land disappear, so the pharaoh's tax collectors had to recalculate each area to adjust the tax amounts. Another need, present in commercial relations, was knowing how to calculate the volume of each grain deposit.

"Measure and Form in Geometry," by Elon Lages Lima, does not require sophisticated knowledge from the reader. The historical curiosities he selected reveal that the determination of areas and volumes is among the first geometric notions that aroused man's interest. His choice to introduce geometry, placing it in the historical context of its appearance, makes the book more fascinating and facilitates the study of the notion of measurement in geometry in one-, two- and three-dimensional aspects, that is, the measurement of line segments (length), plane figures (area) and solid figures (volume). This volume is a modified and enlarged reissue of another book by the author published by the Brazilian Society of Mathematics. Therefore, this fourth edition features new exercises, a new chapter, and a general revision of the text.
22. Problemas e Soluções (1992), by Elon Lages Lima.
22.1. Note.

This book was included in the 2002 edition of Coordenadas no Plano, first published in 1991.
23. Coordenadas no Espaço (1993), by Elon Lages Lima.
23.1. Description of the 2020 edition by the Brazilian Mathematical Society.

In antiquity, linear systems aroused great interest in Oriental mathematics. The Chinese, who were especially fond of diagrams, represented linear systems using coefficients written with bamboo sticks on the squares of a chessboard. Thus they developed the method of resolution by elimination, that is, cancelling coefficients through elementary operations. In the book "Coordenadas no Espaço," by Elon Lages Lima, which is part of the Professor of Mathematics Collection, systems of linear equations are the main theme. In their study, several mathematical theories are inserted, such as, for example, analytic geometry in several dimensions, vector calculus, theory of matrices and determinants.

The author draws attention to the fact that, although it is rarely said, the resolution of linear systems has a geometric interpretation. For example, studying the resolution of a linear system with two equations and two unknowns is the same as looking for intersection of lines in the plane. This is essential for your good understanding. Linear systems are presented to students starting in high school and this book is an excellent reference for both high school and university students because, before introducing the study of this subject, it makes an objective presentation on analytic geometry in the space. The publication also discusses the notions of matrices and determinants, using this theory to develop Cramer's Rule - one of the most traditional methods for solving systems of linear equations - in addition to relating the algebraic notion of determinant to the geometric notion of volume.
24. Algebra Linear (1995), by Elon Lages Lima.
24.1. Description of the 2020 edition by the Brazilian Mathematical Society.
https://loja.sbm.org.br/algebra-linear-exercicios-e-solucoes.html

This book contains the solutions of the exercises proposed by the book 'Álgebra Linear', by Elon Lages Lima, from that same collection.

Each chapter begins with a list of definitions related to it, as well as a summary of the main propositions to be used. Such definitions and summaries precede the excellent solutions, permeated with fine observations, presented by Professor Ralph Teixeira.
25. Isometrias (1996), by Elon Lages Lima.
25.1. Description of the 2007 edition by the Brazilian Mathematical Society.
https://loja.sbm.org.br/isometrias.html

We could paraphrase a famous expression to talk about isometries: "It's elementary, my dear teacher". Although popular when talking about the British literary character Sherlock Holmes, this phrase never appeared in any of the 56 short stories or four books written by Sir Arthur Conan Doyle about the Baker Street detective. In the same way, it cannot be said that for many mathematics teachers it is elementary to work in the classroom with displacements in the straight line, in the plane and in space, which are called isometries in Elon Lages Lima's book.

The author's presentation on this topic is brief. In cases where isometries appear – transformations that preserve the Euclidean distance, Elon seeks to classify them and analyze the composites of these transformations. The book's 12 topics are accompanied by examples and exercises. This helps to understand situations where isometries occur; the use of coordinates on the line, on the plane and in space; the composition of isometries in the plane and in space; and the proper and improper isometries. So, yes, we can say: "It is elementary to know these cases, my dear professor."
26. A Matemática do Ensino Médio (1997), by Elon Lages Lima.
26.1. Description of the 2007 edition by the Brazilian Mathematical Society.
https://loja.sbm.org.br/matematica-e-ensino.html

The reader will find in this work a collection of essays on mathematics for schools up to the 6th grade. The material gathered helps to better understand concepts and topics not explored in detail in basic education. The definition of convex and equidecomposable polygons, classical problems on graphs, quadratic equations, teaching of linear systems, logarithmic systems, and the determinant of the product of two matrices are discussed. The latest essays presented are essential for all teachers interested in education and science communication. In them, the author discusses the teaching of mathematics, the three levels of the discipline in Brazilian schools and the first courses in mathematics in Brazil.

In an interview reproduced in the first chapter, Elon Lages Lima emphasises that any young person in basic education is capable of learning mathematics. Discipline can help students to have more concentration, since the generality with which mathematical propositions are valid requires precision and prohibits ambiguities. It's not about having a talent for understanding mathematics. "Perseverance, dedication and order at work are indispensable qualities for the study of mathematics," he teaches.

26.2. Review by: Paulo Ventura Araújo.
Gazeta de Matemática 148 (2005), 44-46.

Elon Lages Lima is the author of the best mathematics textbooks that, over the last forty years, have been written in Portuguese. His books in the Projecto Euclides and Matemática Universitaria collections, successively republished by IMPA (National Institute of Pure and Applied Mathematics, Rio de Janeiro, Brazil), not only contributed to the learning of generations of students, but also conveyed to them a taste for elegant exposition, lucid and orderly maths topics. It is worth naming some of these books: in the Projecto Euclides collection, the two-volume Analysis Course (the first one already has eleven editions) and the text on Metric Spaces; and, in the Matemática Universitaria collection, the Linear Algebra manual, where, through its exquisite exposition and original perspective, Elon manages to breathe new life into a subject that the profusion of textbooks has trivialised to the point of irrelevance. Both Metric Spaces and Linear Algebra won the Jabuti Prize awarded by the Brazilian Book Chamber for the best science book published in that country in the respective year (1978 and 1996).

As Portugal is not Brazil and, despite the official rhetoric, our cultural and scientific ties are fragile, not many Portuguese students learned mathematics by reading Elon Lages Lima. There weren't many, and today there will be even fewer, because IMPA books are almost never seen in our bookstores. And it's a shame, because those ephemeral manuals that fill the shelves are not like Elon's: some will be honest, even scientifically irreproachable; but whoever writes them is not, as Elon so obviously is, a writer. What difference is there between a writer and a non-writer? The former weighs every word and, instead of taking refuge in neologisms, looks for the appropriate, vernacular name for each new concept; the second thinks that any name will do, as long as the terminology is consistent. What the first calls a "zigzag", the second would call, without blushing, a "z-configuration". The first knows that behind every mathematical convention there is a story that needs to be clarified, even if this lengthens the exposition; the second is all about bare efficiency, the motto that follows the definition and precedes the theorem, all militarily numbered.

To fulfil his vocation as a writer, Elon has written assiduously, always about mathematics. After the university manuals, in recent years he dedicated his attention to secondary education (equivalent to our secondary education), directing an improvement course for teachers that has been running at IMPA since 1990 and launching, as support for the course, a collection of books, Meu Professor de Matemática, which now has 19 titles, 12 of which he authored (alone or in partnership). They are elementary books, which clarify and deepen the curricular themes of mathematics of this teaching level; their value and usefulness, however, far transcend the immediate purpose for which they were written: any professor or student of mathematics, in any country or educational level, could read them with pleasure and profit (if the language were not an obstacle).

The book under consideration, Mathematics and Teaching, jointly edited in Portugal by SPM and Gradiva, originally appeared as the 16th volume of the Meu Professor de Mathematics collection. The spelling and terminology were adapted for the Portuguese edition. The book is divided into 17 short independent chapters. Roughly speaking, chapters 1, 15, 16 and 17 deal with the teaching of mathematics in its various aspects (pedagogical, methodological, social), while the rest deal with specific mathematical themes (synthetic and analytical geometry, combinatorics, linear algebra and analysis).

In the first chapter, Eton interviews himself about teaching mathematics in Brazil. Even though some problems in Brazil have an unparalleled gravity among us, such as miserable salaries and the social disqualification of teachers, there is a question that does not lose relevance: why the teaching of mathematics has, there as here, worse results than the other subjects? There are certainly specific reasons for this poor performance, and Elon lists a few: mathematics is a cumulative knowledge, which does not forgive previous gaps; its teaching requires precision, care, and order; and, finally, its learning requires effort, which goes against the grain of modern pedagogies ("in the past, it [fear of mathematics] was shared with fear of Latin, but this has been abolished, along with almost everything that required work in the school curriculum").

Chapters 15 and 16 are essays where Elon discusses some of the major and disparate trends that have shaped mathematics teaching: so-called modern mathematics with its "conjunctivitis" and over-conceptualisation; the obsessive manipulation disconnected from conceptualisation (a flaw in some current textbooks); the belief in computers as a miraculous remedy to raise teaching to instant modernity.

Chapter 17 reproduces an interview between Elon and Nuno Crato published in 2001 in the weekly Expresso. It talks about the beginning of mathematical research in Brazil, the creation of IMPA, Elon's own personal journey, and finally the teaching of mathematics, with harsh words for the domain of the "sciences" of education, to the detriment of mathematical content, in teacher training.

The properly mathematical subjects, which occupy the largest portion of the book, begin with three chapters on plane geometry: convex polygons (Chapter 2), sum of angles of a polygon (Chapter 3) and equidecomposable polygons Chapter 4). The comparison, in Chapter 2, of several possible characterisations, local or global, of a convex polygon, is very instructive and contains the germ of one of the most fruitful studies of current mathematics, that of convex geometry. In the next chapter, an apparently trivial question, that of calculating the sum of the angles of a polygon, is the pretext for an elegant demonstration that any polygon (convex or not) can be composed of a finite number of juxtaposed triangles. ... Bolyai's (father) famous theorem, that two polygons with the same area are necessarily equidecomposable, occupies Chapter 4; Noteworthy is the inclusion of the refinement of this result by Hadwiger and Glur, who proved that, using only translations and half-turns, any two equidecomposable polygons can be cut out and rearranged so as to overlap each other.

Chapter 5 contains a quick introduction to the graph concept, with the novelty of also using the classic Königsberg bridge problem to explain, in a very suggestive way, the notion of dual graph.

The question "why does space have 3 dimensions?" sets the tone for Chapter 6, which contains a careful reflection on those geometric postulates that, implicitly or explicitly, translate the three-dimensionality of space.

These first mathematical chapters of the book (2 to 6) are at the same level as the best mathematical popularisation literature: that which seduces with its intelligence, honestly explaining how the little toys (fragments or foreshadowings of more sophisticated theories) work that it exposes to the reader, instead of wanting to dazzle him with the narration of great deeds whose essence doesn't give him the slightest clue.

The remaining chapters on mathematical topics (7 to 14) are not of a lower level, but their content and tone are different: they seek to clarify controversies, correct errors and open up new points of view on subjects in the school curriculum that, due to being so often repeated, become they would say, incapable of arousing interest in the experienced reader. But Elon brilliantly overcomes this challenge: who would say that there was something new to tell about the quadratic equation, the division of magnitudes into proportional parts, systems of linear equations, logarithms, linear or exponential growth? Apropos the quadratic equation, arguably "paleontological" subject, Elon, using the convexity of a branch of a hyperbola, provides a visual interpretation of the inequality between arithmetic and geometric means; the same interpretation makes it possible to discuss the number of roots as a function of the coefficients and is the pretext for an introduction to the important iterative method of approximate calculation of roots.

This ability to circumvent the obvious and look at things from a new angle could be said, lightly, that it is like drawing water from a stone. But, because our eyes were tired, we didn't see the source that was hidden behind the stone; With this book, Elon teaches us how to look better.

We hope that such a timely edition, as editions of excellent books always are, will have the success it deserves and will encourage publishers to continue the endeavour.

Porto, 26 September 2004
27. Geometria Analítica e Álgebra Linear (2001), by Elon Lages Lima.
27.1. Description of the 2015 edition by the Brazilian Mathematical Society.
https://loja.sbm.org.br/geometria-analitica-e-algebra-linear.html

The book covers analytic, plane, and spatial geometry. Elon Lima explains that this means, on the one hand, the study of geometry through the introduction of coordinates, and, on the other hand, the method of looking at algebra and analysis problems from the point of view of geometry. It is, therefore, a valuable reference for students who are in their first year of graduation.

In the book, 47 topics are presented, all accompanied by exercises. Among them are the coordinates of the line and of the plane; the distance between two points; the equations of the line; the area of a triangle; vectors in the plane; the equations for the ellipse, hyperbola and parabola; Cramer's rule; and matrices and quadratic forms. Some of the items covered are part of the high school syllabus, but ideally readers will already need to be familiar with them to get the most out of the book.
28. Topologia e Análise no Espaço $\mathbb{R}^{n}$ (2015), by Elon Lages Lima.
28.1. Description by the Brazilian Mathematical Society.
https://loja.sbm.org.br/topologia-e-analise-no-espaco-rn.html

Real number functions, the normed vector space $\mathbb{R}^{n}$ and topological space $\mathbb{R}^{n}$, integral functions, and fundamental theorems of differential and integral calculus are the topics studied in topology in multidimensional Euclidean spaces and analysis. The topics that make up the book's chapters are aimed at students at the end of their undergraduate and beginning master's degrees. The appendices, in addition to containing the solutions to the exercises, address the concept of surface and discuss differential varieties.

Ronaldo de Lima's tip to make the most of this edition is to have knowledge of basic facts about linear algebra and linear analysis. The publication brings the necessary material for students to venture into studies where theories are applied or generalised in fields such as differential geometry, complex analysis, differential equations and general topology.

Last Updated February 2023