# Juan Jorge Schäffer's books

Juan Jorge Schäffer wrote four books. We give some information on these below including some publisher's information and some extracts from reviews.

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Linear differential equations and function spaces (1966) with José Luis Massera

Geometry of spheres in normed spaces (1976)

Basic Language of Mathematics (2014)

Linear Algebra (2014)

1. Linear differential equations and function spaces (1966), by José Luis Massera and Juan Jorge Schäffer.
1.1. The Publisher's Summary of the Chapters.

Chapter 1. Geometry of Banach spaces.

This chapter discusses the geometry of Banach spaces or of a pair of Banach spaces in duality. It discusses various concepts dealing with the apartness of elements and subspaces, which replaces the concept of angle, proper to Euclidean spaces. The chapter is devoted to the geometrical properties of pairs of Banach spaces, "coupled" by means of a bilinear functional; the simple and most important case - essentially the only one if the spaces are reflexive - is the evaluation coupling between a Banach space and its dual. It is often necessary to decide when two subspaces of a Banach space are "close" to one another; in other words, to define a topology, or even a metric, in the set of all subspaces of a Banach space. All vector spaces have as scalar field - occasionally denoted by "F" - either the real or the complex field; the former is denoted by "R."

Chapter 2. Function spaces.

This chapter discusses the several classes of function spaces that play a fundamental role throughout the study of differential equations that follows. The elements of these spaces appear in two distinct capacities: (1) as parts of the equations themselves, and (2) as prospective solutions of the equations. The chapter is restricted to the case of ordinary differential equations, with a real independent variable and only considers spaces of functions whose domain is an interval "J" of the real line "R"; the values of the functions, however, may belong to any Banach space "X."

Chapter 3. Linear differential equations.

This chapter discusses certain differential equations - namely, the linear equations. The independent variable, always called "t", ranges over a subinterval "J" (that is, a connected set containing at least two points) of "R"; the domain J, whether left arbitrary or further specified, remains fixed and generally unmentioned. The chapter also includes some fundamental properties including existence and uniqueness of solutions, formulas that connect the solutions, and bounds for these solutions. The theory of associate equations in coupled spaces, including that of the adjoint equation, and the corresponding Green's Formula, is developed in the chapter, and contains some preparatory material on the set of those solutions of the equations that belong to a given function space.

Chapter 4. Dichotomies.

This chapter discusses the generic term dichotomy, more precisely, (ordinary) dichotomy, and exponential dichotomy, respectively. It describes two types of behaviour of the solutions, which might be termed as "uniform conditional stability," and "uniform asymptotic (or exponential) conditional stability," respectively: uniform, because independent of the "initial" value to; conditional, because some solutions remain (or become) small, while others remain (or become) large. There is in addition an "apartness" condition on the two sets of solutions.

Chapter 5. Admissibility and related concepts.

This chapter discusses and defines admissibility and related concepts. The fundamental theme in the chapter is the relation between certain "test functions" f and nice solutions for f. The crude expression of this theme is the notion of admissibility of a pair of classes of functions, both in $L(X)$. The chapter deals with the concept of admissibility, in the case in which both classes are Banach function spaces of the type described. The fundamental result is a boundedness theorem.

This chapter discusses admissibility and dichotomies. The main structure of the chapter consists of "direct" and "converse" theorems. It also describes certain very neat kinds of behaviour - namely, ordinary and exponential dichotomies - of the solutions of the homogeneous equation and explores direct theorems short of those involving dichotomies. The chapter also discusses the fundamental inequalities.

Chapter 7. Dependence on A.

This chapter discusses the dependency of several mathematical properties on the operator function "A" in a given equation and defines how they vary when "A" varies. The chapter contains a sort of perturbation theory, although some parts of it are more precisely concerned with "roughness" properties of the equations - for example, insensitivity of certain patterns of behaviour to small modifications of "A." The chapter also defines several classes of dichotomy and deals with exponential dichotomies first: they constitute the richer and more interesting case.

Chapter 8. Equations on R.

This chapter discusses the problems and equations in which J = R. The results obtained in this chapter shows that, where the analogue of a theorem for equations on R+ exists, it is in general a straightforward extension through the cut-and-splice method. The chapter describes the relations between function spaces on R and, on R- , R+ . All statements, formulas, and so on, are understood to apply to J = R as the domain of t, unless, the contrary is either explicitly stated or is implied by the following agreement; a subscript - or + refers to restrictions on R- or R+ , respectively; the exact nature of this restriction for several objects are also explained in the chapter.

Chapter 9. Ljapunov's method.

This chapter discusses the information on Ljapunov's method. The concept of a dichotomy, ordinary or exponential of the solutions is closely related to the notion of uniform stability, ordinary or asymptotic, respectively. This suggests the possibility of adapting the so-called Ljapunov's (second or direct) method to the characterisation of dichotomies. The definition of Ljapunov's functions and their total derivatives, and the statement of the relevant assumptions are mentioned in the chapter. The chapter also contains characterisations of exponential and ordinary dichotomies, respectively, by means of the existence of Ljapunov's functions with suitable properties. The order is suggested by the fact that the results and their proofs are simpler, neater, and more comprehensive in the exponential case.

Chapter 10. Equations with almost periodic $A$.

This chapter describes the equations when $A$ is a (uniformly) almost periodic function of $t$; often, it is also assumed that $f$ is almost periodic. The problems considered in this chapter are only to a small extent of periodic equations. Certain extensions to Stepanoff almost periodic functions (related to $M$ as the uniformly almost periodic ones are to $C$) are possible to include all integrable periodic functions. The chapter also assumes that the essential results of the theory of (uniformly) almost periodic functions with values in a Banach space; this theory was developed by Bochner, and it closely parallels the case of real-valued functions. The chapter also considers the modules (additive subgroups) of real numbers, henceforth simply called "Modules" and denoted in general by German letters.

Chapter 11. Equations with periodic $A$.

This chapter describes the equations and existence of periodic solutions when $A$ is periodic. For periodic equations in finite-dimensional $X$, there is availability of well-known canonical reduction to equations with constant $A$, associated with the name of Floquet. However, such a "Floquet representation" is not possible in infinite-dimensional $X$. The chapter also discusses the simple case of the double exponential dichotomy. The case of ordinary double dichotomies is complicated by the possible lack of disjointness of the dihedron.

Chapter 12. Higher-order equations.

This chapter discusses high-order equations. These are first-order equations (although the values of $x$ and $f$ lie in a Banach space $X$) in which only the first derivative of the solution is involved. The chapter remarks that, with a somewhat greater effort at technical refinement, the same methods can be applied to systems of equations with each equation of a different order, yielding essentially the same results.

1.2. Review by Constantin Corduneanu.
Mathematical Reviews MR0212324 (35 #3197).

The book contains in a more systematic and detailed form the main results in the theory of linear differential equations which the authors have obtained in the last years. The theory is developed for equations in a Banach space and considerable emphasis is placed on the methods of functional analysis and on the use of function spaces. The book is addressed primarily to readers interested in the theory of differential equations, but no specialised knowledge in this field is required. ... The book is well-written, self-contained and constitutes undoubtedly an important tool in this field of investigation. The reviewer believes that the topics and methods of this book will find interesting developments in other branches of modern research such as control theory and functional equations.
2. Geometry of spheres in normed spaces (1976), by Juan Jorge Schäffer.
2.1. Review by: Robert C James.
Bull. Amer. Math. Soc. 84 (1) (1978), 70-71.

Geometric properties of the unit sphere of a Banach space have proved to give much information about the general nature of the space. For example, it has long been known that a Banach space is reflexive if its unit sphere is uniformly convex; this has been strengthened, so that it is now known that $X$ is isomorphic to a space for which no two-dimensional sections of the unit sphere are nearly squares if and only if $X$ is super-reflexive (no non-reflexive space has all its finite-dimensional subspaces "nearly isometric" to subspaces of $X$). Another spectacular example is the fact that all infinite-dimensional Banach spaces have arbitrarily large finite-dimensional subspaces that are nearly Euclidean, which has been widely useful and revealing. This book contains much new information about certain aspects of the geometry of unit spheres. It might be described as a detailed and comprehensive study of the girth, perimeter, radius, and diameter of unit spheres of Banach spaces. This field is new and interesting, perhaps even weird. It is not yet clear how important it will be for the study of Banach spaces, but it has connections with several concepts of current research interest, e.g., super-reflexivity, the Radon-Nikodym property, infinite trees, and preduals of $L^{1}(\mu)$-spaces. Although accessible to beginning students, the book seems primarily of value to research mathematicians interested in some of the concepts mentioned in this review. A non-specialist might be confused by the frequent mixing of important and not-so-important facts.

Mathematical Reviews MR0467256 (57 #7120).

In an earlier paper [in 1967], the author initiated the study of some quantities defined in terms of the inner metric of the unit sphere in a normed space. ... This monograph collects the results obtained in this direction over the decade. Some new material is also included and many proofs are improved. However, many topics which in the reviewer's opinion fit the book's title are not even mentioned. The author studies in detail the relations between the parameters and the manner in which the various extrema that appear in their definitions are or are not attained. Special attention is given to the class of flat spaces ...

2.3. Review by: Edward Odell.
American Scientist 65 (5) (1977), 642.

Schaffer's book is concerned with the geometry of unit spheres of normed linear spaces, with respect to the inner metric (the distance between two points in the greatest lower bound of the lengths of all curves connecting the points). It is a useful collection of much of the research in this area in the last ten years. The non-specialist in normed spaces can expect to encounter some difficulty; for example, Chapter 14 is entitled "Girth and Super reflexivity," yet the reader is referred elsewhere for the definition of super-reflexive.
3. Basic Language of Mathematics (2014), by Juan Jorge Schäffer.
3.1. From the Publisher.

This book originates as an essential underlying component of a modern, imaginative three-semester honours program (six undergraduate courses) in Mathematical Studies. In its entirety, it covers Algebra, Geometry and Analysis in One Variable. The book is intended to provide a comprehensive and rigorous account of the concepts of set, mapping, family, order, number (both natural and real), as well as such distinct procedures as proof by induction and recursive definition, and the interaction between these ideas; with attempts at including insightful notes on historic and cultural settings and information on alternative presentations. The work ends with an excursion on infinite sets, principally a discussion of the mathematics of Axiom of Choice and often very useful equivalent statements.

3.2. From the Preface.

This work is intended to provide a presentation (with some carefully chosen alternatives) of essential ingredients of mathematical discourse. It deals with concepts: set mapping, family, order, natural number, real number, finite and infinite sets, countability; with procedures; proof by induction, recursive definition, fixed point theorem, and some immediate applications. It concludes with a sketch of maximality axioms for infinite sets, with the Axiom of Choice and other useful equivalents. The resulting account was designed to support a component of a comprehensive three-semester honours programme, "Mathematical Studies", conducted for years at Carnegie Mellon University, by the author with the inspiring partnership of Walter Noll.

3.3. Contents.

1. Sets.
2. Mappings.
3. Properties of Mappings.
4. Families.
5. Relations.
6. Ordered Sets.
7. Completely Ordered Sets.
8. Induction and Recursion.
9. The Natural Numbers.
10. Finite Sets.
11. Finite Sums.
12. Countable Sets.
13. Some Algebraic Structures.
14. The Real Numbers: Complete Ordered Fields.
15. The Real Number System.
16. The Real Numbers: Existence.
17. Infinite Sets.
4. Linear Algebra (2014), by Juan Jorge Schäffer.
4.1. From the Publisher.

In the spirit of the author's Basic Language of Mathematics, this companion volume is a careful exposition of the concepts and processes of Linear Algebra. It stresses cautious and clear explanations, avoiding reliance on co-ordinates as much as possible, and with special, but not exclusive, attention to the finite-dimensional situation. It is intended to also serve as a conceptual and technical background for use in geometry and analysis as well as other applications.

4.2. Review by: Robert Piziak.
Mathematical Reviews MR3244281.

Linear algebra is surely a core subject in the training of students of mathematics as well as of scientists, engineers and others. The sheer number of books on this subject available today is testimony to this claim. When a new book is presented on the subject, it is legitimate to ask why. Many texts share the same topics in much the same order, as is the case with many calculus books. However, the book under review stands out as quite different from most. The author states in his preface: "This book is a companion volume to the author's Basic language of mathematics and is inspired by the same concern to provide a clear, comprehensive and formally sound presentation of its subject matter." ... [Let us look at] the author's claims in his preface. First, this book is not in the mould of the well-known works of Lay, Leon, Strang, Halmos, Kaplansky, etc. There are no exercises (problem sets) at all and few examples. This book is more than a "companion" to Basic language. It is not stand-alone, but relies totally on Basic language for its notation and proofs to be understood. Is it "clear"? I guess that is in the eyes of the reader. ... proofs are typically referred to previous results and even to results in Basic language. Is it "comprehensive"? In 130 pages, how can that be? ... it should be evident which topics are missing (eigenvalues, determinants, polynomials, ...). Is it "formally sound"? I guess that depends on how formally sound Basic language is. It certainly is formal and notationally dense. On the positive side, the author has done a meticulous job of writing. However, the audience for this book is at best a subset of those who have mastered Basic language.

Last Updated February 2023