1.1. Review by: J Dieudonné.
Mathematical Reviews MR0029470 (10,610c).

After four preliminary paragraphs devoted to the exposition of the essential elementary notions of algebra and topology, the author approaches the theory of topological vector spaces under an extremely general aspect: the field of scalars can indeed be any commutative topological field, and neither the topological vector space nor the field of scalars are assumed to be separate; it gives in particular the characterisation of the neighbourhoods of 0 in a topological vector space, and a very careful exposition of the theory of bounded parts.

2.1. Review by: J Dieudonné.
Mathematical Reviews MR0118802 (22 #9571).

This leisurely exposition of the theory of Haar measure is intended for students with a minimum amount of knowledge of anything beyond calculus (essentially, the first definitions in general topology and the theory of Banach spaces); everything else which is needed (starting, for instance, with compact spaces, topological groups, and semi-continuous functions) is treated in the book itself. This includes in particular the part of integration theory on locally compact spaces which is required for the main results on Haar measure; this turns out to be much less than one would expect, and does not even include Lebesgue's convergence theorem or the general form of the Lebesgue-Fubini theorem; it is developed by the author after the method of Bourbaki. This preliminary material occupies about one half of the book; the rest is concerned with the basic properties of invariant and relatively invariant measures on locally compact groups and their homogeneous spaces; quasi-invariant measures (on homogeneous spaces for which no relatively invariant measure exists) are not considered. The book is well and clearly written, and one of its attractive features is the large number of examples treated by the author as well as many remarks which aim at locating the theory both historically and in relation with neighbouring parts of mathematics. It certainly constitutes one of the best introductions to the subject.

3.1. Review by: J Sebastiao e Silva.
Mathematical Reviews MR0124599 (23 #A1911).

This is a lecture in which the author essentially presents the results of Malgrange and Ehrenpreis on the application of the theory of distributions to partial differential equations.

4.1. Review by: J Korevaar.
Mathematical Reviews MR0213868 (35 #4722).

This book is devoted to distributions on a finite-dimensional real vector space E. Its most notable feature is that the treatment is coordinate-free throughout. The book is essentially self-contained, and it would make a good text for a beginning semester course on distributions, especially if the instructor would supply a two to four week introduction on the one-dimensional case.

The first 60 pages deal with m times and infinitely differentiable mappings between normed vector spaces, culminating in a Taylor's formula for such mappings. The next 70 pages are devoted to the definition and elementary properties of distributions. A distribution on $E$ is defined in the usual way as a continuous linear functional on the space of test functions on $E$, that is, a space of infinitely differentiable (real) functions with compact support. The author discusses distributions of finite order, differentiation, multiplication, and convolution. This part ends with a characterisation of translation invariant continuous linear mappings between spaces of infinitely differentiable functions on $E$ as convolution mappings. The next 60 pages deal with Fourier transformation ... The book ends with a discussion (without proofs) of a form of Schwartz's kernel theorem, the Titchmarsh-Lions support theorem, and the division theorems of Hörmander and Lojasiewicz.

5.1. Comment.

This book is an English translation of the Portuguese book Integral de Haar (1960) which was reviewed by J Dieudonné (see above).

5.2. From the Introduction.

The present volume contains the material of an introductory course on the Haar integral which the author had the opportunity of giving successively, during the second semester of 1959, at the Faculdade Nacional de Filosofia of the University of Brazil (Rio de Janeiro, Guanabara) and the Instituto de Física e Matemática of the University of Recife (Recife, Pernambuco).

5.3. Review by: J C Oxtoby.
The American Mathematical Monthly 73 (8) (1966), 918.

Despite the number of existing books about topological groups there is definitely a place for this one. Addressed to graduate students "having the mathematical maturity normally expected from them," and presupposing only a rudimentary knowledge of general topology, algebra, and elementary integration, its aim is to reach the existence and uniqueness theorems for the Hear integral by the shortest possible path. It is hard to imagine how this aim could be more successfully achieved. The exposition moves smoothly and self-sufficiently through the theory of the Radon integral in a locally compact space, then proceeds to locally compact groups. The main existence and uniqueness theorem is stated and assumed. while properties of the Haar integral are derived from it and abundantly illustrated with examples from the classical groups. Then two proofs are presented, following A Weil and H Cartan, respectively. A short final chapter on locally compact homogeneous spaces reveals the applications of the Haar integral to integral geometry.

The exposition is strictly according to Bourbaki, and the book provides an excellent introduction to this theory of integration. Thus measure plays only a subordinate role; there is no need to distinguish between Baire and Borel measures, and questions of regularity or of the possibility of invariant extensions do not even arise. There are no lists of problems or exercises. One might criticise the book on the ground that it makes the subject appear almost too finished, but within the limits the author has set himself he has done a superb job.

6.1. Review by: R R Phelps.
Mathematical Reviews MR0208228 (34 #8038).

Starting with the definition of a topological vector space, the first third of these notes is devoted to that part of the theory (locally convex spaces, Hahn-Banach and separation theorems) needed in the sequel. The essential content of the remaining material is a series of theorems in approximation theory, that is, theorems which characterise the closures of certain subsets of the space  $\mathscr C (E; K)$ of all continuous $K$-valued functions on the completely regular space $E$, in the topology of uniform convergence on compacta. (Here, $K$ is the real or complex field.) Some of the subsets considered are the following: sublattices (Kakutani-Stone), ideals, subalgebras (Weierstrass-Stone), submodules over a subalgebra (Weierstrass-Stone) and convex sup-lattices (Choquet-Deny). The main part (about half) of the book is devoted to the topic of weighted approximation for submodules of $\mathscr C (E; K)$. This is fairly recent work which generalizes the classical Bernstein approximation problem and which is, for the most part, taken from original work of the author. ... These notes are largely self-contained, the main exception being the Denjoy-Carleman theorem (to which there are four references in the extensive bibliography). The exposition is leisurely and readable, with comments on alternative proofs and viewpoints scattered throughout.

7.1. From the Preface.

In 1948 I published three notes in the Comptes Rendus de l'Académie des Sciences (Paris) containing results on the relationship between topological and order structures. In 1950 I also wrote a note along the same line for the Proceedings of the International Congress of Mathematicians Cambridge, Mass.). Most of these results were developed in a monograph entitled Topologia e ordem which I wrote in Portuguese while I was at the University of Chicago and which was printed by the University of Chicago Press in 1950. This monograph was a thesis that I submitted to the Faculdade Nacional de Filosofia, Universidade do Brasil (Rio de Janeiro), in 1950, as a candidate for a vacant chair in analysis. Many of my results in this direction have been re-obtained and used in the past 15 years, mostly by mathematicians interested in the structure theory of topological semigroups, in the classification of closed semi-algebras of continuous real-valued functions, and in dynamical systems. My Portuguese monograph being inaccessible to these readers, its translation into English is hereby presented.

7.2. Review by: D R Brown.
Mathematical Reviews MR0219042 (36 #2125).

This monograph consists primarily of an English translation of the author's Portuguese tract Topologia e ordem (1950). The contents include an introduction and three chapters. The introduction contains the standard definitions concerning general topology, ordered sets, and vector spaces, as well as a brief but well organised historical outline of the notions of topological spaces and ordered sets. The author indicates his preference for the nomenclature pre-order, order, and total order, replacing, respectively, quasi-order, partial order, and order. ... Since there is no material here of later date than 1950, it follows that this is not in any sense an up-to-date treatise on the marriage of order and topology. However, the reviewer, and any other Portuguese-ignorant person who has wrestled with the earlier edition, should be delighted with an English version, particularly one so clearly translated and well reproduced.

8.1. From the Preface.

This text was mimeographed by the University of Rochester, Rochester, N.Y., in 1964. That edition was very limited and available only through private distribution. A new edition was mimeographed in 1965 [Reviewed by R R Phelps (see 6.1 above)]. After correction of misprints and errors noticed in the Rochester and Rio de Janeiro editions, the text is now reproduced once again.

8.2. Review by: R B Holmes.
Mathematics of Computation 23 (106) (1969), 444-445.

It is appropriate to begin by pointing out that the subject matter of this book is not best approximation; the author is rather concerned with the problem of arbitrarily good approximation. More precisely, the author works within the framework of a given function algebra $C(E)$, consisting of all continuous scalar (real or complex, depending on the circumstances) functions on a completely regular topological space $E$. Such algebras are given the compact-open topology and the general problem is then to characterise the closure of various subsets $S$ of $C(E)$.
...
The book will be accessible to readers with a modest background in analysis (Taylor and Fourier series, Stirling's formula) and general topology (partition of unity, Urysohn's lemma). The necessary functional analysis of locally convex spaces is developed in the early chapters. The Denjoy-Carleman theorem on quasi-analytic functions is the only other major result needed and references for its proof are provided. There is an extensive bibliography, but no index or exercises.

It is clear that numerical analysts will find material on approximation more relevant to their profession in, for example, the books of Cheney or Rice. On the other hand, Nachbin's book provides an interesting blend of hard and soft analysis, and more importantly, it collects together for the first time the main closure theorems in function algebras. For these reasons the book represents an important contribution to the mathematical literature. But it also merits additional kudos: the author is noted for (among other things) the clarity of his mathematical exposition and the present book continues in this trend. It is a pleasure to read!

9.1. From the Preface.

The present report on spaces of holomorphic mappings was prepared for the Sexto Colóquio Brasileiro de Matemática (Poços de Caldas, Minas Gerais, Brazil, July 1967). I also had the opportunity of giving a series of lectures on this material while I was a visiting member at the Center for Theoretical Studies of the University of Miami (Coral Gables, Florida, USA, February 1968). The preparation of this report was sponsored in part by the USA National Science Foundation through a grant to the University of Rochester.

I am glad to thank Professors Paul R Halmos and Peter J Hilton for accepting my text as part of the series Ergebnisse der Mathematik und ihre Grenzgebiete.

9.2. From the Introduction.

The purpose of this monograph is to describe a natural method of endowing certain vector spaces of holomorphic mappings with locally convex topologies, and to derive a few results for the sake of illustration of the simple ideas involved in such a method.

10.1. Review by: M Hervé.
Mathematical Reviews MR0274798 (43 #558).

This book reproduces a course given several times by the author, most recently at the University of Rochester in 1967; its reading is easy and instructive for a beginner. The first chapter deals with the various classical aspects of analytical functions of one or more variables; the second essentially contains the theorem of extension of an analytic function on $U-K$ ($U$ open containing $K$ compact), with a new proof due to Ehrenpreis, and the Cartan-Thullen theorem, with applications; the third is limited to the algebraic properties (theses are local, Noetherian, factorial) of the ring of germs of analytic functions at a given point.

11.1. From the Introduction.

In this book we present a different approach in proving the results contained in our previous paper [L Nachbin, S Machado and J B Prolla, Weighted approximation, vector fibrations and algebras of operators, J. Math. Pures Appl. 50 (1971), 299-323]. These results were concerned with weighted locally convex spaces of cross sections and with algebras of operators. The viewpoint we shall adopt here consists in firstly proving the so-called bounded case of the weighted approximation problem, and then use it to treat the general case. This approach corresponds to the one used in [L Nachbin, Weighted approximation for algebras and modules of continuous functions: Real and self-adjoint case, Ann. of Math. 81 (1965), 289-302] for the case of modules of continuous functions, whereas the approach of our previous paper corresponds to the one used in [L Nachbin, Elements of Approximation Theory, Van Nostrand, Princeton, NJ, 1967].

The weighted spaces of cross sections contain as a particular case the weighted spaces of vector-valued functions. For these it is possible to generalise many of the results about scalar-valued functions which do not generalise to cross sections. For such generalisations see [J B Prolla, The weighted Dieudonné theorem for density in tensor products, Indag. Math. 33 (1971), 170-175], where the weighted Dieudonné theorem for density in tensor products is treated; [J B Prolla, Weighted spaces of vector-valued continuous functions, Ann. Mat. Pura Appl. 89 (1971), 145-158], where the dual of a weighted space of continuous vector-valued functions on a locally compact space is determined; and [J B Prolla, Bishop's generalized Stone-Weierstrass theorem for weighted spaces, Math. Ann. 191 (1971), 283-289], which concerns the non-self-adjoint bounded case of the weighted approximation problem.

11.2. Review by: J P Ferrier.
Mathematical Reviews MR0385573 (52 #6434).

The authors present results contained in a previous paper concerning weighted approximation and its generalisations. The proofs are however different as they first consider the "bounded case" of the weighted approximation problem and use it to treat the general case.

12.1. Review by: Ph Novaerraz.
Mathematical Reviews MR0385564 (52 #6425).

The author introduces a class of locally convex vector spaces where, to study analytic functions, we can reduce ourselves to the case of normed spaces.

13.1. From the Publisher.

Introduction to Functional Analysis: Banach spaces and Differential calculus presents students of pure and applied mathematics with a thorough introduction to Banach spaces and differential calculus on them. The main objective of this text is to show aspects of the increasingly sophisticated uses of functional analytic techniques in mathematics, physics, engineering, economics, statistics, and information theory.

The book is divided into two parts of twelve sections each. The first part is a self-contained introduction to Banach spaces. The second part presents differential calculus in normed spaces, independent of coordinate systems. Special emphasis is placed both on methods and abstract ideas. The geometric descriptions are applicable to finite and infinite dimensions. A feature of the text is the introduction of vector space terminology to the discussion of differential calculus. Numerous exercises and examples accompany the text, and a bibliography is included as a guide to further reading and study of functional analysis and its applications.

Introduction to Functional Analysis: Banach spaces and Differential calculus is for the advanced undergraduate and beginning graduate student of pure and applied mathematics and is equally suited for teaching or self-study. The only prerequisite for successful use of this text is a basic understanding of linear algebra and metric space theory.

about the author ...

Leopoldo Nachbin is Professor of Mathematics at he Universidade Federal do Rio de Janeiro, Brazil, and George Eastman Professor at the University of Rochester, New York. Dr Nachbin received his Ph.D. degree (1947) from the Universidade Federal do Rio de Janeiro. He was the recipient of a number of fellowships including a U.S. State Department Fellowship (1948-1949), a Guggenheim Foundation Fellowship (1949-1950 and 1957-1958), and a Rockefeller Foundation Fellowship (1956-1057). Dr Nachbin has taught at universities in North and South America as well as in Western Europe, and has been an invited lecturer at numerous international conferences. His special research interests focus on analysis, particularly functional analysis, holomorphy, and approximation theory.

13.2. Review by: J Danes.
Mathematical Reviews MR0596227 (82f:58002).

This textbook is an elementary introduction to Banach spaces and differential calculus on them. It is intended for undergraduate and beginning graduate students of mathematics. The book is divided into two parts (Banach spaces, Differential calculus). The second part especially is written very carefully (differentiation of inverse mappings, higher order differentiation, natural identifications for multilinear mappings, etc.). The only shortcoming is the absence of applications, and this should be remedied by the lecturer.

13.3. Review by: Joe Diestel.
The American Mathematical Monthly 90 (8) (1983), 579-580.

Professor Nachbin's Introduction to Functional Analysis: Banach Spaces and Differential Calculus is aimed, remarkably enough, at advanced undergraduate and beginning graduate students from economics, engineering, physics and, perhaps, even mathematics. It aims at acquainting the students with elementary facts about normed linear spaces and the theory of differentiation of functions between such spaces. Though the three basic principles are stated within the text, no serious undertaking is made to demonstrate their power. Perhaps functional analysis has reached the point where the initial contact with the subject can be effective by just acquainting the student with the main terms and most elementary facts. Perhaps this is so. But, quite frankly, it seems that Professor Nachbin sold the subject (and himself) a bit short. Had he only stated the Three Principles and used them at several critical junctures, he might well have written THE classic introduction for the type of audience he was addressing. His writing style is so relaxed and clear he just might have pulled off the missionary accomplishment of twentieth century mathematics: the conversion of a whole horde of economists, engineers, and physicists to functional analysis! By not exhibiting the fundamental principles playing their natural roles in functional analysis, Professor Nachbin has made it more difficult for the serious reader to begin on more earnest study of the subject. With all that one now needs to know, this is unfortunate indeed.