1.1. Review by: H Anzai.
Mathematical Reviews MR0087049 (19,294a).

This is a comprehensive book on functional analysis. In Chapters 1-5 fundamental concepts of functional analysis are introduced, and important concrete functional spaces are illustrated. Naturally these chapters are written in a modern way compared with Banach's book "Théorie des opérations linéaires." Chapter 6 deals with the theory of commutative normed rings of the Russian school, which will be a motivation for the following Chapters 7-10. Chapter 7 is concerned with Riesz's theorem on complete continuous operators in a Banach space. This is proved by the method of Cauchy's integration formula which was used by Gelfand to prove the representation theory of normed rings. In Chapter 8 spectral resolution of bounded normal operators is discussed on the basis of representation theory of normed rings. Typical examples of bounded hermitian and unitary operators are given and their spectral resolutions are illustrated. Chapter 9 deals with von Neumann's works on unbounded hermitian operators, accompanied by typical examples. Chapter 10 is concerned with closed operators on F-type spaces, and especially the canonical decomposition theorem of closed operators in a Hilbert space is proved. In Chapter 11 general notions on non-commutative normed rings are explained and their representations in a ring of linear operators on a Banach space are discussed. Again as applications of commutative normed rings Wiener's Tauberian theorem and other important theorems of Fourier analysis are proved. In Chapter 12 the author develops his theory of one-parameter semigroups of linear operators on a Banach space. From this point of view he shows P Lévy's infinitely divisible laws, and the integrability of Fokker-Planck's equations. Chapter 13 deals with vector lattices. Many important theorems on functional analysis are viewed from this stand point. Finally a quick sketch of L Schwartz's distribution theory is given in the appendix.

2.1. Review by: A E Heins.
Mathematical Reviews MR0118869 (22 #9638).

We have here a modern introductory monograph on the theory of ordinary differential equations, together with those aspects of integral equations which are required for the former. There are five main chapters, whose headings are self explanatory. These are: 1. The initial value problem for ordinary differential equations; 2. The boundary value problem for linear differential equations; 3. Fredholm integral equations; 4. Volterra integral equations; and 5. The general expansion theorem (Weyl-Stone-Titchmarsh-Kodaira's theorem). The book closes with some comments on non-linear integral equations, and an appendix which contains several theorems of a more advanced nature from the theory of functions of a complex variable. It is the author's intention to take us through the basic development of the initial value and boundary value problems (regular as well as singular) of ordinary differential equations, and a very readable account is given. The book can be read with profit by a student who has a thorough grounding in the elements of the real and complex calculus minus Lebesgue integration. Of special value are the many examples which are given to illustrate the theory as well as to fulfil the author's desire to let the reader know why one proceeds in a particular direction or why a theory fails and modifications are necessary.

2.2. Review by: William H Pell.
Amer. Math. Monthly 69 (5) (1962), 451-452.

This excellent little book, a translation from the Japanese, is essentially a self-contained introduction to the theory of ordinary differential equations and integral equations. Since both of these subjects are considered, and since the author chooses to give proofs of many of the theorems which he uses, the material covered is not as extensive as that usually found in textbooks on these subjects individually. In particular, applications are not dealt with, except insofar as one may regard examples involving the Legendre, Bessel, Laguerre, and Hermite functions as such. The author's pace is steady, however, and rather brisk, so that he covers sufficient material for a viable introductory survey of his chosen topics.

3.1. Review by: M Fukamiya.
Mathematical Reviews MR0133678 (24 #A3504).

The aim of this book, in Japanese, is to give an easily readable account and survey of the theory and applications of distributions on $\mathbb{R}^{n}$ in the sense of L Schwartz, not only for mathematicians, but also for physicists and engineers, by avoiding the device of locally convex topology. ... Though most of the material can be found, e.g., in the book of Schwartz, there are very readable accounts and notable additions by the author, and this text should be helpful to those who wish either to read the work of Schwartz or to find suitable references on distributions.

4.1. From the Preface.

The present book is based on lectures given by the author at the University of Tokyo during the past ten years. It is intended as a textbook to be studied by students on their own or to be used in a course on Functional Analysis, i.e., the general theory of linear operators in function spaces together with salient features of its application to diverse fields of modern and classical analysis.

Necessary prerequisites for the reading of this book are summarized, with or without proof, in Chapter 0 under titles; Set Theory, Topological Spaces, Measure Spaces and Linear Spaces. Then, starting with the chapter on Semi-norms, a general theory of Banach and Hilbert spaces is presented in connection with the theory of generalized functions of S L Sobolev and L Schwartz. While the book is primarily addressed to graduate students, it is hoped it might prove useful to research mathematicians, both pure and applied. The reader may pass, e.g., from Chapter IX (Analytical Theory of Semi-groups) directly to Chapter XIII (Ergodic Theory and Diffusion Theory) and to Chapter XIV (Integration of the Equation of Evolution). Such materials as "Weak Topologies and Duality in Locally Convex Spaces" and "Nuclear Spaces" are presented in the form of the appendices to Chapter V and Chapter X, respectively. These might be skipped for the first reading by those who are interested rather in the application of linear operators.

In the preparation of the present book, the author has received valuable advice and criticism from many friends. Especially, Mrs K Hille has kindly read through the manuscript as well as the galley and page proofs. Without her painstaking help, this book could not have been printed in the present style in the language which was not spoken to the author in the cradle. The author owes very much to his old friends, Professor E Hille and Professor S Kakutani of Yale University and Professor R S Phillips of Stanford University for the chance to stay in their universities in 1962, which enabled him to polish the greater part of the manuscript of this book, availing himself of their valuable advice. Professor S Ito and Dr H Komatsu of the University of Tokyo kindly assisted the author in reading various parts of the galley proof, correcting errors and improving the presentation. To all of them, the author expresses his warmest gratitude.

Thanks are also due to Professor F K Schmidt of Heidelberg University and to Professor T Kato of the University of California at Berkeley who constantly encouraged the author to write up the present book. Finally, the author wishes to express his appreciation to Springer-Verlag for their most efficient handling of the publication of this book.

Tokyo, September 1964

4.2. Review by: Ivan Singer.
Mathematical Reviews MR0180824 (31 #5054).

The book comprises 15 chapters: (O) Preliminaries; (I) Semi-norms; (II) Applications of the Baire-Hausdorff theorem; (III) The orthogonal projection and F Riesz' representation theorem; (IV) The Hahn-Banach theorems; (V) Strong convergence and weak convergence; appendix to Chapter V: Weak topologies and duality in locally convex linear topological spaces; (VI) Fourier transform and differential equations; (VII) Dual operators; (VIII) Resolvent and spectrum; (IX) Analytic theory of semi-groups; (X) Compact operators; appendix to Chapter X: The nuclear space of A Grothendieck; (XI) Normed rings and spectral representation; (XII) Other representation theorems in linear spaces; (XIII) Ergodic theory and diffusion theory; (XIV) The integration of the equation of evolution. The book also contains a Bibliography including about 300 items and an Index (of subjects and authors). As shown by the above table of contents, the book covers a great variety of topics in functional analysis. Besides, each chapter of the book contains a great deal of material, both classical and modern, as well as many useful examples in concrete spaces. Of course, the book also includes the results of the author, who has enriched functional analysis during the last 30 years with many valuable contributions. Actually, each chapter of the book deserves to be treated in a separate monograph and, in fact, as is well known, for most of them there do already exist such monographs. This allows the author to omit deliberately in each chapter several topics, e.g., in the theory of normed linear spaces, series, projections, tensor products, homeomorphism problems, approximation of compact operators by operators of finite rank, invariant subspaces, etc. are not mentioned.

5.1. From the Preface.

In the preparation of this edition, the author is indebted to Mr Floret of Heidelberg who kindly did the task of enlarging the Index to make the book more useful. The errors in the second printing are corrected thanks to the remarks of many friends. In order to make the book more up-to-date, Section 4 of Chapter XIV has been rewritten entirely for this new edition.

Tokyo, September 1967

6.1. From the Preface.

A new Section (9. Abstract Potential Operators and Semi-groups) pertaining to G Hunt's theory of potentials is inserted in Chapter XIII of this edition. The errors in the second edition are corrected thanks to kind remarks of many friends, especially of Mr Klaus-Dieter Bierstedt.

Kyoto, April 1971

7.1. From the Preface.

Two new Sections "6. Non-linear Evolution Equations I (The Komura-Kato Approach)" and "7. Non-linear Evolution Equations 2 (The Approach Through The Crandall-Liggett Convergence Theorem)" are added to the last Chapter XIV of this edition. The author is grateful to Professor Y Komura for his careful reading of the manuscript.

Tokyo, April 1974

7.2. Review by: G D.
Giornale degli Economisti e Annali di Economia, Nuova Serie 34 (11/12) (1975), 798.

This is a volume that not only proves the exceptional development and the wide success of studies on linear operators in function spaces that are today observable in Japan, from which country there have also been important examples of modern times in the field of mathematical economics (although there its results have not yet become known abroad), but also the progressive refinement made after the first edition of 1964 by the well-known mathematician. The transformations or functions, also called mappings, in the topological spaces of Hilbert and Banach constitute the most debated and essential logical notions of the book. These are also forms of argumentation employed and dealt with increasing frequency as well as by mathematical researchers of the "exact sciences", also by theoretical economists when they intend to specify with rigour and deepen traditional economic theories with this type of methodological approach. The book is particularly well-known for its clarity of exposition, which lends itself especially to support some of the economic practices currently in vogue.

8.1. From the Preface.

Taking advantage of this opportunity, supplementary notes are added at the end of this new edition and additional references to books have been entered in the bibliography. The notes are divided into two categories. The first category comprises two topics: the one is concerned with the time reversibility of Markov processes with invariant measures, and the other is concerned with the uniqueness of the solution of time dependent linear evolution equations. The second category comprises those lists of recently published books dealing respectively with Sobolev Spaces, Trace Operators or Generalized Boundary Values, Distributions and Hyperfunctions, Contraction Operators in Hilbert Spaces, Choquet's Refinement of the Krein-Milman Theorem and Linear as well as Nonlinear Evolution Equations.

A number of minor errors and a serious one on page 459 in the fourth edition have been corrected. The author wishes to thank many friends who kindly brought these errors to his attention.

Kamakura, August 1977

9.1. From the Preface.

Two major changes are made to this edition. The first is the rewriting of the Chapter VI, 6 to give a simplified presentation of Mikusinski's Operational Calculus in such a way that this presentation does not appeal to Titchmarsh's theorem. The second is the rewriting of the Lemma together with its Proof in the Chapter XII, 5 concerning the Representation of Vector Lattices. This rewriting is motivated by a letter of Professor E Coimbra of Universidad Nova de Lisboa kindly suggesting the author's careless phrasing in the above Lemma of the preceding edition. A number of misprints in the fifth edition have been corrected thanks to kind remarks of many friends.

Kamakura, June 1980

9.2. Publisher's Information (on 1988 reprint).

The present book is based on lectures given by the author at the University of Tokyo during the past ten years. It is intended as a textbook to be studied by students on their own or to be used in a course on Functional Analysis, i.e., the general theory of linear operators in function spaces together with salient features of its application to diverse fields of modern and classical analysis. Necessary prerequisites for the reading of this book are summarized, with or without proof, in Chapter 0 under titles: Set Theory, Topological Spaces, Measure Spaces and Linear Spaces. Then, starting with the chapter on Semi-norms, a general theory of Banach and Hilbert spaces is presented in connection with the theory of generalized functions of S L Sobolev and L Schwartz. While the book is primarily addressed to graduate students, it is hoped it might prove useful to research mathematicians, both pure and applied. The reader may pass, e.g., from Chapter IX (Analytical Theory of Semi-groups) directly to Chapter XIII (Ergodic Theory and Diffusion Theory) and to Chapter XIV (Integration of the Equation of Evolution). Such materials as "Weak Topologies and Duality in Locally Convex Spaces" and "Nuclear Spaces" are presented in the form of the appendices to Chapter V and Chapter X, respectively. These might be skipped for the first reading by those who are interested rather in the application of linear operators.

10.1. Review by: H G Dales and D G Crighton.
SIAM Review 28 (1) (1986), 98-99.

It is, of course, a programme of some antiquity to seek to solve differential equations by reducing them to algebraic equations. One method for this is to use Laplace transformations, and a well-known and powerful technique for implementing this method is Mikusinski's operational calculus. The aim of the present work is "to give a simplified exposition as well as an extension of" this calculus. This is surely a worthwhile task. ... It is disappointing that the applications given in the second half of the book are only the standard ones, and are accessible to other treatments. They include the one-dimensional wave equation, the telegraph equation and the heat equation. The calculations are carefully done. ... In summary, we feel that there is a clear and important idea struggling to emerge in this book, but that it is in the end suffocated in tedious calculations and a desire, not really sustainable, to write at a very elementary level.