**1. Functional analysis and applied mathematics (1952), by L V Kantorovich.**

**1.1. Review by: H P Thielman.**

The object of the present work, according to the author, is to show that the ideas and methods of functional analysis can be used for the development of effective, practical algorithms for the explicit solutions of practical problems with just as much success as that with which they have been used for the theoretical study of these problems. He further states that in some cases the explicit results obtained from the more general point of view prove to be more complete and precise than those obtained for special cases of the problem.

**2. Approximate Methods of Higher Analysis (1958), by L V Kantorovich and V I Krylov.**

**2.1. Review by: T J Rivlin.**

*SIAM Review* **2** (4) (1960), 299.

This volume (previously known to those ignorant of Russian in a German translation) is a most welcome addition to the literature in English on computational methods in mathematical physics. Because of its thoroughness and scope it is by far the best reference book on the subject. In the manner of many Russian texts the authors have not hesitated to provide background material at the expense of lengthening the book. This has the effect of making the book eminently readable and accessible to both students and practitioners of calculation. Another good quality of the book is the authors' habit of giving examples which are worked out in complete detail.

**2.2. Review by: Philip Rabinowitz.**

*Science, New Series* **134** (3487) (1961), 1358.

this book by Kantorovich and Krylov, originally titled Methods for the Approximate Solution of Partial Differential Equations, ... has been expanded to include material on integral equations and conformal mapping. The volume contains much material not available elsewhere, especially results of the Russian school of numerical analysis. It also contains many numerical examples, which unfortunately do not indicate the scope of the methods illustrated since they are intended to illustrate hand calculations. ... the richness of the book will provide many new ideas for the interested reader

**2.3. Review by: Hans Sagan.**

*Amer. Math. Monthly* **67** (5) (1960), 485.

On the jacket, the book is praised as a "unique text and reference book of pure and applied mathematics". It can be considered as an almost unique text because of the wealth of information contained therein and the great number of carefully worked out examples, many of which are not trivial.

**2.4. Review by: Edward M Wright.**

*The Mathematical Gazette* **44** (348) (1960), 145.

This book is concerned with approximate methods used in the solution of partial differential equations, conformal mapping and the approximate solution of integral equations. ... The translation ... occasionally has a literal flavour. None of these trivial criticisms however detract from the very great usefulness of a translation of a standard text in a field to which Russian mathematicians have contributed largely.

**2.5. Review by: Alston S Householder.**

*Mathematics of Computation* **14** (69) (1960), 90-91.

The book itself is concerned mainly with the numerical solution of partial differential equations, as the title to the first edition (1936) indicated. The first chapter deals with expansion in series, both orthogonal and nonorthogonal, with a section on the improvement of convergence. Next come methods of solution of Fredholm integral equations with applications to the Dirichlet problem. Then comes a chapter on difference methods, and one on variational methods. This accounts for slightly more than half of the book. There follow two chapters, for a total of nearly 250 pages, on conformal methods, and finally about 50 pages on Schwarz's method. Throughout, the presentation is extremely readable, with the inclusion of numerous examples, but no exercises.

**2.6. Review by: W F Freiberger.**

This work is a translation from the third Russian edition [1950], which was a minor revision of earlier editions. The work is concerned with what Lanczos calls 'parexic analysis', a study of processes which lead to approximate solutions of the problems of mathematical physics by rigorous methods. This is in contradistinction to 'numerical analysis' which deals with the translation of mathematical processes into numerical algorisms, and is greatly aided by the discoveries of parexic analysis. In other words, the work under review presents a survey of analytical tools which are of importance in numerical analysis, with emphasis on error estimation.

**Economic Calculation of the Optimum Use of Resources (Russian) (1959), by L V Kantorovich.**

**3.1. Review by: J M Montias.**

*Econometrica* **29** (2) (1961), 252-254.

This recent book is far wider in scope than any of the author's previous publications. While it contains a number of examples drawn from the practice of individual enterprises, it also stresses economy-wide problems and the general principles of efficient allocation: at a point of time (static model), through space (transportation problem), and through time (dynamic model). He lays particular emphasis on the use of efficient prices, derived from the solution of a linear program, to bring about marginal improvements in resource allocation, without the need to resort to a recomputation of the entire program. He claims that the implicit prices are sufficiently stable-given small changes in the demand for the output and in the set of attainable points-to justify this sort of approximation.

**3.2. Review by: Robert W Campbell.**

*The American Economic Review ***50** (4) (1960), 729-731.

In this book Kantorovich has developed his ideas far beyond the point achieved in the original article. He now recognizes the broader significance of his resolving multipliers and acknowledges their meaning by labelling them "objectively determined values." Much of his book is devoted to the demonstration that the finding of these values is equivalent to finding the optimum plan, that is the plan which gives the largest output. In the course of the exposition he develops the concepts of opportunity costs, the rental value of superior land, quasirents on capital goods, and scarcity values for current inputs.

**4. Tables for the numerical solution of boundary value problems of the theory of harmonic functions (1963), by L V Kantorovich, V I Krylov and K Ye Chernin.**

**4.1. Review by: M G Arsove.**

An approximate solution of the Dirichlet problem for a fixed region with arbitrarily prescribed boundary values [is] be obtained from tabulated values of [certain] coefficient functions ... The present work provides such tables for the following regions: (1) rectangle, (2) half-plane, (3) strip, (4) halfstrip, (5) angle, (6) circle, (7) half-circle. Similar tables are included for the normal derivative and for the second boundary-value problem (Neumann problem) (the latter only for the case of a rectangle).

**5. Functional Analysis in Normed Spaces (1964), by L V Kantorovich and G P Akilov.**

**5.1. Review of the 1959 Russian edition by: Edwin Hewitt.**

This treatise is designed for readers familiar with the theory of functions of a real variable and with linear algebra: there are no other formal prerequisites. The authors are particularly concerned with applications of functional analysis to the theory of approximation and the theory of existence and uniqueness of solutions of differential and integral equations (both linear and non-linear). This point of view has led them to include a great many specific examples of spaces and operators, some not readily accessible elsewhere, many involving intricate computations, and many of considerable interest. This emphasis on concrete applications is welcome and should make the book of wide usefulness.

**5.2. Review by: J H Webb.**

*The Mathematical Gazette* **50** (374) (1966), 438-440.

This hefty book gives a systematic account of the theory of normed spaces, the theory of linear operators on these spaces and the theory of functional equations. The book is divided into two parts, "Linear Operations and Functionals" and "Functional Equations". ... At every stage great stress is laid on applications. But there are no exercises for the reader, who is seldom trusted to prove anything himself. The encyclopaedic nature of this book, and its high price, destine it to become a work of reference rather than a classroom text. ... This is a good book, well written and well translated.

**5.3. Review by: John J Sopka.**

*SIAM Review* **11** (3) (1969), 412-413.

According to the authors: "The book is based on a course of lectures . for students specializing in mathematical analysis and computational mathematics." As such its contents are more oriented toward real analysis than is the case in many of the more recent texts on normed and/or topological linear spaces. The material and style of presentation are, nevertheless, within the domain of abstract analysis. ... this is a book which can be recommended for the reference shelves of a rather broad class of mathematicians.

**The Best Use of Economic Resources (1965), by L V Kantorovich.**

**6.1. Nemchinov writes in the Preface.**

With the level of development of the national economy and the exceptional complexity of internal economic relations, the problem of finding the best possible system of planning would become insurmountable without a radical improvement in methods of economic calculation and the utilization of the latest computing techniques. The use of modern mathematical methods in the organization and planning of production provides a real and very efficient method of improvement. It is therefore not surprising that linear programming as an independent discipline first emerged in the Soviet Union. Important results in this field were achieved in 1938-9 by the author of this book, L V Kantorovich, and published by him in a number of works beginning in 1939. The first of these contained fundamental advances and determined the content and further development of this discipline: it examined the mathematically new type of "extremal" problems; it evolved a universal method for their solution (method of solution multipliers) as well as various efficient numerical algorithms derived from it; it indicated the more important fields of technical-economic problems where these methods could be most usefully applied; and it brought out the economic significance of indicators resulting from an analysis of problems by this method which is particularly essential in problems of a socialist economy.

**6.2. Review by: Maurice Dobb.**

*Science & Society ***31** (2) (1967), 186-202.

The author is the distinguished Russian mathematician who invented what is now known (both East and West) as linear programming: an Academician and a recipient last year (along with Novozhilov and Nemchinov) of a Lenin Prize.

**6.3. Review by: Ralph W Pfouts.**

*Annals of the American Academy of Political and Social Science* **361** (1965), 168-169.

L V Kantorovich, a Russian mathematician, developed the concept and some of the theory and methods of linear programming in 1939, several years before American economists and mathematicians published anything on the subject. The present book, which was published in the Soviet Union in 1959, includes a great deal of material relating to practical problems of socialist economic planning as well as material relating to the theory of linear programming. Indeed, the bulk of the book is made up of three lengthy chapters on problems of socialist planning. The first chapter deals with simple cases of short-run planning of production. The second chapter deals with more complicated problems of short-run planning, while the third chapter deals with problems of long-run planning or problems of expanding the stock of capital.

**6.4. Review by: Robert Dorfman.**

*The American Economic Review* **56** (3) (1966), 592-597.

The English translation of The Best Use of Economic Resources was eagerly awaited by bourgeois economists for a number of reasons. Economists with a special interest in programming and production theory wanted to honour the discoverer of linear programming by reading his own presentation of his discovery and to gain the insight that only the words of an original thinker can convey. Students of trends in Soviet economic thinking wanted to study at first-hand the position of Academician Kantorovich, winner of the Lenin Medal for meritorious scientific achievement and most eminent exponent of formal, mathematical methods of planning. The second group of students will find in, and particularly between, the lines of this book many valuable clues to the state of current economic thinking in the Soviet Union; the first group, those interested primarily in the economic significance of linear programming, will be seriously disappointed.

**6.5. Review by: Leif Johansen.**

*The Economic Journal* **77** (305) (1967), 123-124.

The main text of the book uses numerical examples for presenting the ideas and drawing general conclusions, while most of the underlying mathematics is given in appendices. The presentation is in most cases clear and efficient, although there are quite a few overlappings and repetitions. The examples are imaginary, but it is my feeling that they reflect in a simplified way important practical problems.

**6.6. Review by: P Göran Ennerfelt.**

*The Swedish Journal of Economics* **67** (2) (1965), 167-171.

The author is known as the father of linear programming. He developed, independently and unnoticed even by Soviet fellow economists, the essential techniques of linear programming already towards the end of the 1930's, whereas in the West this technique was not developed and applied till under and, especially, after the second world war. Kantorovich is in his book mainly concerned with the best use of economic resources in a very wide meaning of these words. The starting point in the book is a description of the irrationalities of the present Soviet economic system. ... Kantorovich uses an interesting approach to the problems of economic valuation and rationality in a centrally planned economy, an approach which in theory is quite acceptable to a Western economist. The discussion in academic circles in the Soviet Union around the proposals of Kantorovich has shown that his solution to some is quite unthinkable and unmarxist, while others with a knack for higher mathematics and less Marx tend to support some of his main propositions.

**7. Essays in Optimal Planning (1977), by L V Kantorovich.**

**7.1. Review by: Alfred Zauberman.**

*Economica, New Series* **44** (176) (1977), 427-428.

The core of Kantorovich's theory is of course the dual revealed at optimum: the Lagrangean multipliers - in his terminology the "objectively conditioned valuations" ("000"s). His stress on objectivity, one feels, was intended to disarm the traditionalist critics sensitive to the sin of "subjectivism". However, it took nearly two decades for Kantorovich's conceptions to obtain recognition in the economic thinking of his own country. By now it has routed the established doctrine; and it dominates the price theory. Also, it is largely thanks to Kantorovich's subsequent elaborations of his original construct that the snags in the application of the optimal price have become understood.

**8. Functional Analysis (1977, 1982, 1984), by L V Kantorovich and G P Akilov.**

**8.1. Review of 2nd edition by: Robert G Bartle. **

*American Scientist* **71** (5) (1983), 546-547.

The first Russian edition of this well known book appeared in 1959 under the title 'Functional Analysis in Normed Spaces'; the English translation was published in 1964. The second Russian edition was published in 1982 with the Russian equivalent of Functional Analysis as a title. Although a number of revisions and additions have been made, the overall flavour of the original book remains. It gives a balanced and comprehensive treatment of the main aspects of functional analysis. The present edition is a definite improvement over the popular first edition and should be welcome to many readers.

**8.2. Review of 2nd edition by: Frank Smithies.**

The first edition of this book appeared in 1959 under another title [*Functional analysis in normed spaces* (Russian)]. In this new edition the general framework has been retained, but extensive revisions have been made, especially in the earlier chapters. The theory of topological vector spaces is now taken as fundamental, instead of being supplementary to the theory of normed spaces; hence the change in the book's title. ... The book continues to be an excellent source of information on the treatment of linear and nonlinear problems of many kinds, and on the specific properties of particular spaces and operators.

**9. Problems in the efficient use and development of transportation (Russian) (1989), by L V Kantorovich.**

**9.1. Review by: Marlies Borchardt.**

The present monograph of Kantorovich, Soviet Nobel prize winner of 1975 and one of the originators of linear programming, is devoted to problems in the efficient use and development of transportation. The book was written from 1976 until 1986 under Kantorovich. After his death in 1986 it was completed by the economists Livshits, Paenson and Tikhomirov. The book is strongly oriented toward questions on the economics of the planning system of his country.