1.1. Comment by the author.

The purpose of this book is to pick up theoretical points in the book of S Banach: Théorie des opérations linéaire, and to arrange them by modern method. I made a course of lectures on Banach spaces at Tokyo University during 1947-48 and had a great mind to write this book. I finished the manuscript in 1947.

1.2. From the Preface.

Functional analysis, namely analysis in infinite-dimensional spaces, changed its feature by the introduction of the notion of semi-order, and spectral theory got an important part in it, as the integration theory did in the classical analysis.

Spectral theory was established first about Hermitian operators in Hilbert spaces. S W P Steen attempted to modernise this spectral theory in Hilbert spaces, and constructed a spectral theory in semi-ordered rings. On the other hand, F Riesz attempted the generalisation of Lebesgue's decomposition in the integration theory, which was established already for totally additive measures, to finitely additive measures, and obtained a new spectral theory about functionals. This spectral theory was modernised by H Freudenthal as one in semi-ordered linear spaces. Modernisation of these two different spectral theories was attempted independently by S W P Steen and H Freudenthal and they established it at the same time in 1936. Furthermore we found that these two different spectral theories fell into the same contents by modernisation.

The spectral theory in semi-ordered linear spaces constitutes the subject of the present volume. The modernised spectral theory indicated just now will be called the first spectral theory, which is constructed by means of Stieltjes integral, considering every spectrum as a continuous one. On the contrary there is another type of spectral theory established by the author, which is constructed by means of Riemann integral in topological spaces, considering every spectrum as a point spectrum. This spectral theory will be called the second spectral theory in the text. The first spectral theory will occupy Chapter II. However, after Chapter III, we shall be concerned only with the second spectral theory, which enables us to develop the theory further.

Totality of bounded continuous functions on a compact Hausdorff space constitutes obviously a semi-ordered ring. Characterisation of this semi-ordered ring was considered by many mathematicians. This problem will be discussed precisely in Chapter VI, applying the second spectral theory.

Now I wish to say a few words about the structure of the book. It embodies the greater part of a coarse of lectures delivered by the author at Tokyo University during 1946-47. The reader needs only to be acquainted with the elementary part of classical analysis. I attempted at first to get it published in Japanese, but did not succeed on account of the present circumstance of Japan. Then I have translated it into English, hoping it to be printed in U.S.A. I have recognised however that it is also impossible in U.S.A. Accordingly I have resolved to publish it by photographic process. I think this book will be a good introduction to the volume I of Tokyo Mathematical Book Series.

1.3. Review by: R S Phillips.
Mathematical Reviews MR0038564 (12,419f).

The principal aim of this book is the systematic presentation of various representation theories for lattice ordered linear spaces and rings. The author has brought together a great deal of material on this subject, much of which stems from his own research. To a certain extent, however, the treatment is over-systematised since practically the entire development is based on the notion of a projector; nowhere is the equally useful notion of a lattice ideal introduced. ... The presentation is quite formal and contains no applications.

2.1. From the Preface.

Functional analysis, namely analysis in infinite-dimensional spaces, had been studied so far in two different parallel ways: Hilbert spaces and Banach spaces. In Hilbert spaces we are concerned with spectral theory or linear operators, and in Banach spaces we considered relations between elements and linear functionals. But we had nothing other than algebraic methods in Banach spaces. Thus it was intended to introduce an analytical method as spectral theory into Banach spaces. A chance for it was given in 1936.

H Freudenthal constructed a spectral theory of elements in semi-ordered linear spaces: Teilweise geordnete Modul, Proc. Akad. Amsterdam 39 (1936), and S W P Steen considered spectral theory in semi-ordered ring: an introduction to the theory of operators (I), Proc. London Math. Soc. (2) 41 (1936). On the other hand, L Kantorovitch discussed convergence relations, introducing semi-order into Banach spaces, namely in normed semi-ordered linear spaces. Normed semi-ordered linear spaces are not so general as Banach spaces, but include all concrete Banach spaces, for instance l_p-space, m-space, L_p-space and M-space. These facts may be considered to show theoretical approach of spectral theory and Banach spaces.

At this point or view I intended first to make arrangement and reform of spectral theory in semi-ordered linear spaces. Some results obtained during 1938-40 were written in a paper: Teilweise geordnete algebra, Japanese Jour. Math. 17 (1941). I had then discovered a new type of spectral theory: eine Spektraltheorie, Prop. Phys.-Math. Soc. Japan 23 (1941), which was constructed by Riemann integral considering every spectrum as a point spectrum, while the previous spectral theory was constructed by Stieltjes integral considering every spectrum as a continuous spectrum. I discussed further normed semi-ordered linear spaces, defining conjugate spaces in a new way: stetige lineare Funktionale auf dem teilweisegeordneten Modul, Jour. Fac. Sci. Univ. Tokyo 4 (1942).

I feel first in 1941 that normed semi-ordered linear spaces are too wide to apply spectral theory, and intend to discover a new notion of spaces which might be stronger but would include all concrete Banach spaces and permit complete application of spectral theory. After research during 1941-47 I could obtain a complete form of modulared semi-ordered linear spaces which seem to be most suitable to this purpose, and I had written a paper: modulared semi-ordered linear spaces. In this paper I had stated fundamental properties of modulared semi-ordered linear spaces with many properties of semi-ordered linear spaces which I had obtained during the war and could not publish. More precise properties and its applications should naturally rely upon future research, but I can believe from fundamental properties that modulated semi-ordered linear spaces will play an important part in future mathematics.

I submitted this paper to the Annals of Mathematics, but it was refused to accept by reason of its length. The manuscript has been kept since August 1949 in the library of the Institute for Advanced Study at Princeton to be at the disposal of the many mathematicians who come to visit the Institute. A copy kept by me has been available for students in Tokyo University. Recently great damage has been done to it. Accordingly I have made a resolution to publish it by photographic process against financial stress.

2.2. Review by: R S Phillips.
Mathematical Reviews MR0038565 (12,420a).

In order to obtain a more detailed theory for universally continuous semi-ordered linear spaces R, the author has introduced the notion of a modular and investigated the properties of such modulared spaces. ... he first half of the book consists of preliminary material much of which is to be found in the book reviewed above. In the second half, the author first classifies modulars by a great variety of properties.

3.1. Review by: J Dieudonné.
Mathematical Reviews MR0046560 (13,753b).

This book is divided into two parts. The first (chap. I to V) develops in a classical way the general theory of topological spaces, uniform spaces and metric spaces. Some results on the extension to a uniform space of a uniformly continuous function defined in a closed part of the space, seem new. The second part (chap. VI to XII) is devoted to topological vector spaces on the field of real numbers.

4.1. From the Preface.

Spectral theory was considered first by D Hilbert about bounded symmetric operators on a sequence space. J von Neumann made this spectral theory take a new form. He generated this theory to unbounded symmetric operators, defining abstractly the Hilbert space. In this new style spectral theory was discussed by many mathematicians: F Riesz, M H Stone, etc.

On the other hand a spectral theory was born in semi-ordered linear spaces as an abstraction of the spectral theory in the Hilbert space. This spectral theory is constructed by means of generated Stieltjes integral, considering every spectrum as a continuous one. On the contrary there is another type of spectral theory established by the author, which is constructed by means of generated Riemann integral in topological spaces considering every spectrum as a point one. This new spectral theory is explained precisely in Volume II of this Series, Modern Spectral Theory (Tokyo, 1950), in the name of the second spectral theory. The author has remarked in an earlier paper, 'Eine Spektraltheorie', Proc. Phys.-Math. Soc. Japan 23 (1941), 485-511, that this spectral theory also is applicable to the Hilbert space, but it is not well known. The purpose of this book is to make this new spectral theory revive in the Hilbert space.

In this text spectral theory is constructed firstly about elements of the Hilbert space, and as its application, we consider the spectral theory of the normal operators. Furthermore, by means of this new spectral theory, we can discuss not only the mean ergodic theorem obtained by J von Neumann, but the individual ergodic theorem obtained by J D Birkhoff, as a problem in the Hilbert space. Readers will be more interested, if they read this text, comparing Volume II of this Series.

This book was written during 1950-51 as Volume IV of Tokyo Mathematical Book Series. But financial difficulties made it impossible to publish this book. Recently I have obtained financial help from Education Ministry of Japan, and it has become possible to publish this book. I express my warmest thanks to those who have made effort to publish this book.

4.2. Review by: Paul R Halmos.
Mathematical Reviews MR0058874 (15,440d).

This book is a highly individualistic account of the known facts (and some of their generalisations) concerning operators on Hilbert spaces. The author's rugged individualism manifests itself not only in his occasionally novel approach, but also in his insistence on his personal and rather unusual terminology. Thus, for example, the cardinal number of a set is called its "density'', by an "ideal'' in a Boolean ring the author means a dual ideal, a spectral measure becomes a "spectrality'', and the interior of a set in a topological space is called its "opener''. Such terminology, when combined with a difficult expository style and frequent violations of idiomatic English usage, results in a book that is very hard to read.
...
The two aspects of his presentation (realisation and dualisation) constitute the principal novelty in the book. A specialist in the field might profit from studying the book in detail and observing the systematic application of techniques that are known but not yet trite.

5.1. Review by: Editors.
Mathematical Reviews MR0073124 (17,387b).

This is a reprint by photo-offset of twenty-one papers by the author (in one case a joint paper with I Halperin) on partially ordered linear spaces. ... The author states in a preface that he felt impelled to give wider circulation to these papers since many of the results, particularly of the earlier papers, were being rediscovered by others.

6.1. Note.

This is an unaltered reprinting of the preliminary part (Sections 1 through 35) of the author's Modulared semi-ordered linear spaces (Maruzen, Tokyo, 1955).

6.2. From the Publisher.

Although there is continuing interest in the general theory of linear lattices, much of Professor Nakano's basic work on the subject has long been out of print. Linear Lattices has been published to facilitate further study. The book reprints the preliminary part of Modulared Semi-Ordered Spaces (Tokyo, 1950) and contains Professor Nakano's discussion of projectors, spectral representation theory, and reflexive and normed space ...

6.3. Review by: M Satyanarayana.
The Mathematics Teacher 61 (7) (1968), 726.

This book is mainly intended for specialists working in spectral theory. It might be of interest to know how the concepts of continuity and convergence in our classical analysis might be extended to spaces other than the space of real numbers. The author considers the semi-ordered linear spaces, and with the topology induced by the order he introduces concepts such as limits, convergence, and different types of continuous spaces (namely, universal, totally, and super-universal spaces). Following the pattern of normed linear spaces, he develops a theory of projectors, projections, and continuous linear-functionals and proves the existence theorems in terms of structure of the base spaces. The manipulations worked out in this book are very valuable to those interested in the impact of order on linear spaces.

7.1. From the Publisher.

A topological group is a special case of a transformation group on a topological space. This book successfully generalises the theory of topological groups to transformation groups and develops a new field of mathematics.

In the first three chapters the theory of uniform spaces is developed, including many new theorems for application to transformation groups. Usually the metric plays an important role in the theory of uniform spaces. But it is omitted in this book because fundamental Theorem 3 makes it unnecessary to develop the theory.

In Chapter IV, for a transformation group on a uniform space, the adjoint uniformity is defined on the same space and the relations between these two uniformities are discussed.

In Chapter V, for a transformation group on a uniform space, the induced uniformity is defined on the transformation group and the case of transitive transformation groups is discussed.

In Chapter VI the invariant measure by a transformation group is considered. Generally there is no invariant measure, and many conditions were considered by many mathematicians. In this book a simple and beautiful condition is obtained by uniformities. In addition, this book is the first to give a condition for the uniqueness. Of course this theory includes the Haar measure on topological groups as a special case.

In the last chapter almost periodic functions are defined by a transformation group. The almost periodic functions on a group were first defined by J von Neumann by taking the Bochner theorem as the definition. In this book the original Bohr definition is generalised by uniformities, and the Bochner theorem also is generalised as a theorem. In addition, the Bohr mean value theorem is proved in the general case. Almost periodic functions also are defined by a transformation group, and a generalisation of the theory of Fourier series is attempted.

7.2. Review by: S N Hudson.
Mathematical Reviews MR0238990 (39 #350).

... the author gives generalisations of standard theorems in locally compact groups to the case of "locally totally bounded transformation groups".  ... The only weaknesses of the book are a lack of motivation as to why uniformities, invariant integrals (or measures), characters, and almost periodic functions are important, and a lack of references to other research in the area of invariant integrals. If the book is to be valuable to scholars, as claimed by the author, whether they be beginning or seasoned, an effort should be made to relate this presentation to the existing body of knowledge - certainly an effort much greater than a mention of Weil and von Neumann in two paragraphs on page ix of the preface. ... The style of the book is that of a numbered sequence of definitions, lemmas, and theorems. There are no examples or illustrations of specific uniform spaces and transformation groups.

7.3. Review by: P.
Amer. Math. Monthly 76 (4) (1969), 434.

This is an exposition of the author's original generalisation of the theory of topological groups to transformation groups on uniform spaces.