Hidegoro Nakano (1909–1974) – on the centenary of his birth 1 ...

Proceedings of the International Symposium on Banach and Function Spaces III (Kitakyushu, Japan, 2009), pp. 99–171, 2011

Hidegoro Nakano (1909–1974) – on the centenary of his birth Lech Maligranda

Abstract.

The life and work of the Japanese mathematician Hidegoro Nakano (1909–1974) is presented. He graduated from the Department of Mathematics, Faculty of Science, Imperial University of Tokyo in 1933 and from 1935 he was a professor at National First High School. In 1936 he got his doctor degree of science and in 1942 he became an assistant professor at the Imperial University of Tokyo. From 1952 he continued to be a professor in Hokkaido University. From 1960 he was a guest professor at the Queen’s University and from 1961 to 1974 he worked at Wayne University in Detroit, USA. He passed away there in 1974. This biography includes his personal data, scientific achievements, a list of books and the list of published papers. He is mostly remembered in mathematics for Nakano spaces and modular spaces but also several of his results are known in the theory of vector lattices (Riesz spaces) and the operator theory in Hilbert spaces. He was the first one who introduced the notion of a modular.

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Nakano’s life and work

Hidegoro Nakano was born on 16 May 1909 in Tokyo, as the first son of Katsugoro Nakano and Kame Nakano. His father came from a family of artisans specializing in Japanese lacquer-finished ware called “urushi”. Received: October 27, 2010 2000 subject classification: Primary 46E30, 46E40, 46A80, 46B42, 46B20, 46E45, Secondary 47A35, 54E15, 01A27 Key words and phrases: Banach spaces, function spaces, Nakano spaces, OrliczNakano spaces, modular spaces, locally solid Riesz spaces, topological Riesz spaces, order continuous functionals, operators on Hilbert spaces, Hahn-Banach theorem, ergodic theorems, group theory

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Photo 1. Hidegoro Nakano and his signature

Katsugoro was trained to carry on the family business. His ancestors lived in Edo (Tokyo) and served the samurai-class of Tokugawa shogunate, making ornamental parts of swords. When such a profession became obsolete in the 20th century, by applying his skills with paint brushes, he pioneered in business of painting commercial signs. Hidegoro had two sisters and one brother. He was the oldest son. He finished the secondary-level school, Shiba-chu (chu for “chu gakko”, i.e., middle school) in Tokyo and went to National First High School (Ichiko for short), preparatory school for the Imperial University of Tokyo. Its education level corresponds to that of the first two years of general education curricula of today’s 4-year colleges/universities. Hidegoro Nakano studied at the Imperial University of Tokyo and in March 1933 he obtained the degree of bachelor of science. He was a student of Takuji Yoshie (1874–1947). In the period 1 April 1933–29 April 1935 he was doing research at the Graduate School of the Imperial University of Tokyo. In October 1935 he got married and in 1936 he has got degree of doctor of science in differential equations at the Imperial University of Tokyo. Kentaro Yano (1912–1993) wrote in [Ya84] the following about the

Hidegoro Nakano

Photo 2. Hidegoro Nakano in high-school uniform. He was either at graduation from Shiba-chu or entrance to Ichi-ko

doctor degree of Nakano: In 1931 I began my study at the Department of Mathematics, Faculty of Science, Imperial University of Tokyo. At that time, it was 3-year education system at the university. The first year, second year and third year were called primary stage, middle stage, and last stage, respectively. When I was a student at the primary stage, I heard that there was an ingenious student at the middle stage whose name was Hidegoro Nakano. He specialized in the theory of differential equations. In this area, he has already written out his creative and original thesis. When I was a student at the last stage, he had already been in the Graduate School. Especially, on the day when he became a research student, he went to his supervisor’s office and applied for doctor degree by presenting his thesis which was already done. Here let me explain the situations at that time. It is always true that the doctor degree is awarded only after the presented thesis is examined and approved. But at that time it was common that a person was allowed to present his thesis only after he had been doing research for about 20 or 30 years and he reached a suitable age. But Nakano had just graduated from the university, and presented his thesis for applying doctor degree on the same day when he was in the

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Photo 3. Hidegoro Nakano during mid 50s

graduate school. This also astonished his supervisor, Professor T. Yoshie. After a while Nakano told me why he did so. The reason was that there was no age limit for presenting doctor thesis. And according to the regulations, there was no examination fee charged if a person was a student in the graduate school. After all it was a special case indeed. Therefore, Professor T. Yoshie examined Nakano’s thesis very carefully. He arrived at the conclusion that the author of the thesis was well qualified to be awarded a doctor title. In 1936, Nakano was awarded doctor of science degree successfully. At that year, he was 27-year old and close to 28-year old. This nearly 28-year doctor of science was reported a lot in the newspaper. The reason was that the people who got doctor degree were normally at least over 50-year of age at that time. This was really a big news and there was more to say later. Some people who were older than Nakano kept on doing research and had published many papers. But they thought that they were not old enough to apply for doctor degree. Nakano’s case made them think: if Nakano who was younger than them could make it, why they could not do, too. Therefore, there came in the world a lot of younger doctors who were a bit more that 30-year old. And this is also benefi-

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cial to me. This made it possible for me to get my doctor of science degree before the age of 30. Nakano’s motto on PhD was (cf. Ando [An09], p. 1): PhD is a passport to the research community. After PhD, Nakano’s research interest went to modern areas of Hilbert space operators and Banach lattices. In the period of 30 April 1935 – 31 March 1938, Nakano was a professor at Ichi-ko (The National First High School). Then, in the period April 1938 – August 1939, he made research at home and in the years 1939–1952 he worked at the Imperial University of Tokyo being first a research assistant (30 Sept. 1939–30 March 1940), then an instructor (31 March 1940–14 April 1942) and an assistant professor (15 April 1942–31 March 1952). In 1952 he moved to Sapporo, where he became a professor at Hokkaido University (1 April 1952–31 January 1961).

Photo 4. Sapporo May 5, 1957. Hidegoro Nakano

Kentaro Yano [Ya84] wrote the following about Nakano’s definition of professor: Nakano was an assistant professor at the University of Tokyo until 1952, then he was a professor at Hokkaido University.

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L. Maligranda Like a feudal lord, a professor can do whatever he likes without any intervention from others. I imagine that Nakano was happy to be able to give lectures and to do research as he liked. Among the words which were said to be spoken by Nakano and reached to my ears was the following: “A professor is a person who delivers the theory he created himself. A person who retails the theory created by others cannot be called a professor.” Though I was yet an “assistant” professor, there might be many “professors” whom Nakano’s words touched on their sore spots. Though I am not skilled in English, the word “professor” may originate from the word “profess” which means “clearly mention” or “confess belief ”, from this point the correct definition of the word “professor” may be just what Nakano mentioned.

Photo 5. Spring 1959. Sapporo. First row: Sumiko Nakano. Second row from the left: Richard Brauer (Harvard University), Nakano, Akitsugu Kawaguchi (Hokkaido University)1 , Mrs. Kawaguchi, Mrs. Brauer 1

Richard Brauer (1901–1977), Akitsugu Kawaguchi (1902–1984).

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In meantime he was a visiting professor at the Queen’s University in Canada (1960–1961). He visited Israel Halperin (1911–2007), who was working there from 1939 to 1966. They had published 2 joint papers already in 1951 and 1953. Moreover, in 1957 Amemiya, a student of Nakano, was one year visitor of Halperin at Queen’s University. In the letter from 26 April 1960 Nakano informed Professor W. Orlicz about the following: Knowing that a conference of functional analysis will be held in your country in September, I desire to attend this conference but I cannot afford to go.2 About this point, I wrote already to Professor Mazur, but do not get any answer yet. (. . . ) I am going to Canada, being invited by Canadian Mathematical Congress. I am scheduled to leave Sapporo May 26, and to arrive at Vancouver June 23. I shall stay in Kingston for about a year.

Photo 6. Spring 1961. Campus of Queen’s University. Faculty members of the Department of Mathematics, including visitors. Hidegoro Nakano is the 3rd person from the right in the front row 2

The conference in Functional Analysis was in JabÃlonna-Poland on 4–10 September 1960.

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From February 1961 to March 1974 Nakano was a professor at Wayne State University in Detroit, USA. Daniel Waterman in reminiscences on his academic life has written about Wayne State University and here Nakano’s name also appeared (cf. [Wa08], p. 4): Wayne was a stimulating environment for Nishiura and me. It had a strong group in analysis, including Vladimir Seidel, Frederick Bagemihl, Hidegoro Nakano, Albert BharuchaReid, Takashi Ito, and Leon Brown. The graduate students were also very strong. It seemed that Detroit had several gifted students who, for various reasons, were unable to leave the area. We also had some very good foreign students. Let us say some more about Nakano’s family. On 24 October 1935 he got married to Sumiko (11 December 1913 in Tokyo–5 March 1999 in Detroit) and they had daughter Kazumi Nakano (born on 21 June 1937 in Tokyo) and son Hideaki Nakano (born on 7 April 1943 in Tokyo). Both children were educated in Japan. They came to USA in 1962 to join their parents in Detroit. They still live in the USA. Nakano has no grandchildren. An important part of Nakano’s activity was his doctoral students. Nakano was a supervisor of 16 students who got doctor of science or Ph.D. [8 all at Hokkaido University in Sapporo, Japan and 8 at Wayne State University in Detroit, USA]: In old system (Kyu-sei): • 1954 Ichiro Amemiya (1923–1995), A general spectral theory in semi-linear spaces • 1954 Sadayuki Yamamuro (1925), Exponents of modulared semiordered linear spaces • 1960 Shozo Koshi (1928–2003), Approximately additive modulars • 1960 Takashi Ito (1926), On conjugately similar transformations Under the new system (Shin-sei): • 1958 Tsuyoshi Ando (1932), Positive linear operators in semiordered linear spaces

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Photo 7. Nara April 4, 1939. “Mikasa-Yama” park: Hidegoro and Sumiko Nakano with their daughter Kazumi

Photo 8. Winter 1943–1944, home in Tokyo: Hidegoro Nakano and his son Hideaki

• 1958 Tetsuya Shimogaki (1932–1971), On certain property of the norms by modulars • 1959 Masahumi Sasaki (1929), On some properties of modular convergence • 1960 Jyun Ishii (1929–2002), On the finiteness of modulared spaces

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Photo 9. Summer 1962. Sapporo. First row from the left: Amemiya, Nakano, Koshi. Second row from the left: Ito, Ando, Shimogaki

In USA: • 1966 Fredrick W. Barber, Lattice functions of a Boolean algebra • 1966 Richard C. Metzler, On linear topological spaces • 1968 Hans-Heinrich Wolfgang Herda, Modularized spaces of generalized variation • 1971 Bernard C. Anderson, On the converse of the BochnerPhillips theorem in the theory of linear lattices • 1971 Ronald G. Mosier (1938–2008), Discrete elements of the dual of a linear lattice • 1973 Horace Hwa-Tai Chuang, Linear functionals bounded on a lattice basis • 1973 Stephen Romberger, The proper space of a discernible linear lattice • 1975 Joseph Edward Brierly, Representation space of the cluster algebra (completed under Prof. Takashi Ito)

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Kazumi Nakano gave me, in the letter dated 8 June 2008, the list of Ph. D. students of her father. She also informed me about representative members of “Nakano Seminar (Nakano Kenkyu-shitsu)”: Nakano Kenkyu-shitsu existed from 1951 to 1959 while Hidegoro Nakano was an active faculty member of the Department of Mathematics at Hokkaido University. It was a group of graduate students engaged in research under Nakano’s leadership. The group held a weekly seminar in the room next to his office on the third floor of Science Building where the Department of Mathematics was located. Doctoral degrees were conferred to most of them but some left the university without completing the degree requirements. Source of information: Collected works of Nakano Seminar (Nakano Kenkyu-shitsu Ronbun-shu), 1951–1959, three volumes. Department of Mathematics, Hokkaido University. In the group at the Nakano Seminar were above mentioned PhD students and also Tuyoshi Mori, Osamu Takenouchi and Koji Honda. In the next letter from 19 November 2008 Kazumi Nakano explained more about Nakano’s students in Japan: In the old education system of Japan prior to the end of world war II, graduate studies in Mathematics involved very little work of scheduled classroom lectures. Seminars were dominant. Much of the 3-year curricula for undergraduate mathematics majors was similar. Admission to the graduate program was more like doctoral candidacy today. Regular members of Nakano’s seminars, undergraduate as well as graduate, called themselves Nakano’s students. Takenouchi was one of them. Nakano moved to Hokkaido University from Tokyo University in 1952. Both universities were part of the system of the 7 Imperial Universities, i.e., the only universities established and funded entirely by the central government of Japan. An appointment to a position from another institution of the system was regarded as transfer at the discretion of the president of an accepting institution. Nakano’s

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L. Maligranda appointment at Hokkaido University was thus a move from Tokyo to Hokkaido. Three students of Nakano, that is Amemiya, Yamamuro and Mori, followed Nakano and became graduate students at Hokkaido University. Mori left Hokkaido without a doctoral degree. Amemiya was the first student of Nakano to receive a doctoral degree. Six of 8 doctoral recipients were appointed in the faculty of Hokkaido University at some point of their lives. Sasaki left the field of Mathematics after received the doctoral degree and became an information scientist.

Photo 10. Hidegoro Nakano in the end of the 60ties (from the Halmos book [Ha87], picture 257)

WÃladysÃlaw Orlicz met Nakano for the first time in Stockholm at the International Congress of Mathematicians (12–22 August 1962) and has

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written on 10 October 1962, in the report from the Congress that his conversations with Prof. B. Jessen (Copenhagen) and Prof. H. Nakano (Hokkaido) were very useful: I was waiting to meet Nakano as the founder of his school of mathematics in Hokkaido and as competitor with the Pozna´ n group about modular spaces. Nakano died on 11 March 1974 in Detroit. His daughter Kazumi Nakano, in the letter from 20 September 2006, informed me the following: Hidegoro Nakano was suddenly hospitalized in Detroit on August 27, 1973, for a surgery of colon cancer. He experienced a series of complications after the surgery and continued to stay in the same hospital until he died on March 11, 1974. I found Nakano’s name at the 80th Annual Meeting of AMS in San Francisco, 15–19 January 1974 with 20 min. talk Correct set theory in session on Set Theory and the Axiom of Choice. Hidegoro Nakano was scheduled to give a talk in this session and the abstract [Na74] was already received before his hospitalization. Due to the illness, Kazumi Nakano made a presentation on his behalf at the meeting.

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Nakano’s research and books

Nakano’s areas in mathematics were (in the historical order): differential equations (1932–1935), operators in Hilbert spaces (1939–1961), normed and Banach spaces (1939–1974), representation theory (1941– 1961), spectral theory (1941–1973), ordered spaces and vector lattices (Riesz spaces) (1941–1974), measure theory (1943–1969), modular vector lattices and modular spaces (1947–1974), ergodic theory (1948– 1950), uniform spaces (1951–1968) and set theory (1966–1974). Nakano published 8 books in English: 1. 2. 3. 4.

Modulared Semi-Ordered Linear Spaces (1950) [Nb8], Modern Spectral Theory (1950) [Nb9], Topology and Linear Topological Spaces (1951) [Nb10], Spectral Theory in the Hilbert Space (1953) [Nb11],

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5. Semi-Ordered Linear Spaces (1955) [Nb13], 6. Linear Lattices (1966) [Nb18] (this is a reprint of the preliminary part of 1), 7. Uniform Spaces and Transformation Groups (1968) [Nb19], 8. Set Theory (1978) [Nb20]. He has written also 12 monographs and text-books in Japanese language, and 94 scientific papers in a period of 43 years (1932–1974). His co-authors from 1951 to 1974 were: Israel Halperin, Tuyosi Mori, Ichiro Amemiya, Michiyo Miyakawa, Masahumi Sasaki, H. W. Ellis, Leon Brown, Richard Metzler, Bertram J. Eisenstadt, Bernard C. Anderson, Stephen Romberger, Joseph E. Brierly, Ronald G. Mosier, Horace H. Chuang and Kazumi Nakano. In the period 1958–1973 he was also a reviewer for Mathematical Reviews. Nakano already in the forties worked in the theory of Banach spaces and topological vector spaces. He was familiar with the Banach book [Ba32]. In the preface to his book [Nb10] he wrote in November 1951 the following: The purpose of this book is to pick up theoretical points in the book of S. Banach: Th´eorie des op´erations lin´eaire, 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. Talking about his monographs we can describe their value by quoting some reviews from the Mathematical Reviews, where reviewers are writing about his books the following opinions: R. S. Phillips in MR0038564(12,419f) about the book Modern Spectral Theory [Nb9]: 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. (...) The presentation is quite formal and contains no applications.

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R. S. Phillips in MR0038565(12,420a) about the book Modulared Semi-Ordered Linear Spaces [Nb8]: 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.

Photo 11. Around 1960. First row from the left: Ichiro Amemiya, Hidegoro Nakano, his wife–Sumiko Nakano, Toshio Shibata. Second row from the left: Jyun Ishii, Osamu Takenouchi, Shozo Koshi, Takashi Ito. Third row from the left: Takeshi Sumitomo, Akio Tsuruga, Tsuyoshi Ando, Tetsuhiro Shimizu, Tetsuya Shimogaki, Hisaharu Umegaki

P. R. Halmos in MR0058874(15,440d) about the book Spectral Theory in the Hilbert Space [Nb11]: This book is a highly individualistic account of the known facts (and some of their generalizations) concerning operators on Hilbert spaces. The author’s rugged individualism manifests itself not only in his occasionally novel approach,

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L. Maligranda but also in his insistence on his personal and rather unusual terminology. (...) 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.

Let us mention that in the book [Nb11] it was announcement that the book of Nakano, Topological Measures will be published by Maruzen but as I can see this book was not published. H. Nakano himself in MR0073124(17, 387b) about the book SemiOrdered Linear Spaces [Nb13]: This is a reprint by photo-offset of twenty-one papers by the author. 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. Ando was very impressed by the original ideas of Nakano in his book Semi-Ordered Linear Spaces [Nb13]. He says, however, that (cf. Nakazi [Na93], p. 2): it is quite regretable that Professor Nakano used his original notations in all of the books, which prevented the writings as widely accepted in mathematics world as they should have been. Ando’s lifelong interest in order structure was coming from Nakano, but as he said (cf. Nakazi [Na93], p. 2): there was little personal contact with the professor because of Nakano’s aristocratism. S. N. Hudson in MR0238990(39#350) about the book Uniform spaces and transformation groups [Nb19]: 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

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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. (...) There are no examples or illustrations of specific uniform spaces and transformation groups. Kentaro Yano informed in [Ya84] also about Nakano’s special symbols in mathematics, which were the main problems to understand his results: For an ordinary mathematician like me, research in mathematics means to read as many papers of other mathematicians as possible and to get some idea from them to write his own papers. Therefore most of mathematical symbols appearing in my papers are those other mathematicians commonly used. A genius like Nakano, however, seems to think first deeply by himself before reading others’ papers. Encountering with a new concept, Nakano necessarily denotes it by his own symbol. Consequently, if many other mathematicians encounter with the same concept and if, unhappily, there already exists a commonly used symbol fit to this concept, Nakano’s symbols sometimes become his own peculiar symbols. In this sense there were opinions that Nakano’s papers are difficult to read. But once I met a mathematician whose research area was the same as Nakano’s. He told me “Nakano has something”. (I think those are words of praise for Nakano’s originality). Concerning the peculiar symbols of Nakano, let me mention the following story. The letters and symbols we Japanese mathematicians use in writing formulas are all western (European) origin. Nakano, however, thought that it might be allowed to use simple Chinese characters for symbols. He defined the upper and the lower functions of a function ϕ and denoted them by ϕ⊥ and ϕ> , respectively. When Professor Claude Chevalley (1909–1984) of Columbia University, French-born mathematician, once visited Japan

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L. Maligranda and gave lectures, he intentionally adopted those ideas of Nakano to please Japanese mathematicians.

In connection to Nakano’s book Set Theory and the Continuum Hypothesis let us present the following four informations: 1. J. Musielak in the letter to W. Orlicz from June 27, 1966 informed that he visited Nakano in Detroit and that Nakano is looking for an editor who can print a new edition of his book on modulars. Moreover, he is planning to publish at AMS monograph Set Theory. This book is connected with the Continuum Hypothesis. As Nakano pronounce, it is another, more general system of axioms of set theory, which he invented. 2. Nakano himself in the letter to W. Orlicz on 7 January 1969 wrote the following: I wrote a book, Set Theory, and submitted it to the American Mathematical Society to have it published in the Colloquium Series. However, it was rejected because of unfavorable reports from two referees. I also submitted it to many publishers, but all of them rejected it because it was original. Some publishers said that this book should be published in a Journal. These circumstances made me decide to give publicity to the important results obtained in this book piece by piece at Meetings of the American Mathematical Society. Nakano then published in 1966–1970 his 8 abstracts [Na66], [Na67], [Na69a], [Na69b], [Na69c], [Na69d], [Na70a], [Na70b] and in 1970 two papers [N70], [N70a]. His book Set Theory was posthumously published in 1978 by his students [Nb20]. 3. K. Yano informed in [Ya84] about Nakano’s Set Theory book in the following way: Thesis or Book? Have been a professor at Hokkaido University for 10 years, Nakano went to Queen’s University to be a guest professor. 1961, he was a professor in Wayne University, Detroit, in Michigan State. It can be said that he was invited as a star professor.

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Photo 12. Photo of Nakano from his book [Nb20] with his short CV (in Japanese). The picture was taken when he was living in Detroit.

As to me, I have no courage at all to challenge the already established theory again. Only the genius, like Nakano, has such a great courage to dare challenge such a theory. He made such a challenge to the famous Set Theory, and wrote a manuscript with title “Set Theory” in English. But when he presented it to the society for publication, it was rejected by saying that as it was not a paper but a monograph. He was advised to find a publishing company for its publication. When he brought it, however, to a publishing company, it was rejected again by saying that this was not a book but a paper to be published in a journal. Back and forth, such that, it took a long time for his creative work to be published. In 1974, he passed away in Detroit. In 1978, after 4-year later, this book was finally published under the efforts of his pupils and followers. In the last paragraph of preface, Nakano wrote “I had been wondering about the existence of spaces which are neither finite or infinite. During the Christ-

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L. Maligranda mas season of 1965, I got an idea of mathematical set theory and I was able to give a solution not only to this problem but also to the general continuum problem. This was truly a gift from God.”

4. J. Brierly (20 July 2004) wrote the following: (. . . ) the famous Continuum Problem falls into this category (strinker problem). Cohen supposedly proved that this is so. For some reason, I doubt Cohen’s proof. Possibly, because my advisor, the late great Hidegoro Nakano, claimed to resolve this problem many years ago. I never knew Nakano to be inaccurate about anything. He was an incredible mathematical mind. I never researched his set theory. But I suspect that somewhere in the mindst of it is a clarification of the Continuum Hypothesis. Some day I will check.

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Nakano’s name in mathematics

Hidegoro Nakano’s name in mathematics appeared in connection to Nakano spaces, Orlicz-Nakano spaces, modulars, Riesz spaces: 3a. Nakano sequence and function spaces For sequences p = (pn ), w = (wn ) with 1 ≤ pn < ∞ and wn > 0 for all n ∈ N let l{pn } (w) denote the space of all real or complex sequences x = (xn ) for which the norm (the Luxemburg-Nakano norm) kxkp,w = inf{λ > 0 :

∞ X n=1

µ wn

|xn | λ

¶pn ≤ 1}

is finite. This space, where the index set N is replaced even by an arbitrary set was considered by Halperin and Nakano [HN53] in 1953. Special cases appeared earlier: wn = 1 Orlicz [Or31] and wn = p1n Nakano [N51b]. These spaces have got name the Nakano spaces l{pn } but they should be called Orlicz spaces.

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THEOREM 1 (Nakano, 1951). (a) The spaces l{pn } and l{qn } pn qn P |pn −qn | < ∞ for some α > 0 (with are equivalent3 if and only if ∞ n=1 α the convention α∞ = 0). (b) If limn→∞ pn = 1, then l{pn } has the Schur property, that is, every weakly convergent sequence is strongly convergent. 1 EXAMPLE 1 (Nakano, 1951). For pn = 1 + log(log(n+4)) we have

that limn→∞ pn = 1, that is, l{pn } has the Schur property and it is not pn P∞ 1 since for every α > 0 we have pn −1 = order isomorphic to l α n=1 P∞ log α = ∞. n=1 α(log(n + 4)) Nakano spaces were investigated by several mathematicians. Let us mention some results (we assume 1 ≤ pn < ∞ if there is nothing stated additionally): •

(Halperin-Nakano 1953): The Nakano space l{pn } has the Schur property ⇐⇒ limn→∞ pn = 1.

•

(Klee 1964): For 1 < pn < ∞ the Nakano space l{pn } has the Banach-Saks property4 ⇐⇒ 1 < inf n∈N pn ≤ supn∈N pn < ∞.

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(Waterman-Ito-Barber-Ratti 1969): The Nakano space l{pn } is reflexive ⇐⇒ 1 < lim inf n→∞ pn ≤ lim supn→∞ pn < ∞.

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(Sundaresan 1971): If 1 < inf n∈N pn ≤ supn∈N pn < ∞, then the Nakano space l{pn } is uniformly convex.

•

(Woo 1973, Skorik 1975) (a) No infinite-dimensional subspace of 0 the Nakano space l{pn } is isomorphic to a subspace of l{pn } if and only if Lim(pn ) ∩ Lim(p0n ) = ∅, where Lim(pn ) denotes the set of all limit points (accumulation points) of the sequence (pn ) in [1, ∞]. (b) If X is an infinite dimensional subspace of the Nakano space l{pn } , then X contains a subspace isomorphic to lp for some p ∈ Lim(pn ).

3 Two sequence spaces X and Y are equivalent if there exists a sequence {un } such that, putting yn = un xn n ∈ N, we obtain a one-to-one correspondence between x = (xn ) ∈ X and y = (yn ) ∈ Y . 4 A Banach space X is said to have the Banach-Saks property if every weakly null sequence in X, say (xn ), contains a subsequence first arithmetical means ‚ P (xnk ) whose ‚ ‚ m ‚ converge strongly to zero, that is, limm→∞ ‚ k=1 xnk ‚/m = 0.

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•

(Skorik 1980) If pn 6= 2 for all n (excluding possibly one of them), then the group of isometries of the Nakano space l{pn } onto itself is the same as for co or lp , p 6= 2, i.e, it consists all of sign alternations and permutations of coordinates.

•

(Kami´ nska 1986): The Nakano space l{pn } is uniformly convex ⇐⇒ 1 < lim inf n→∞ pn ≤ lim supn→∞ pn < ∞ and pn = 1 for at most one index n.

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(Wnuk 1991): The Nakano space l(pn ) has the Dunford-Pettis property5 if and only if Lim(pn ) ⊂ {1, ∞}.

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(Peirats-Ruiz 1992): Let (pn ) be a sequence of positive real numbers such that 0 < inf n∈N pn ≤ supn∈N pn < ∞. Then lp is isomorphic to a (complemented) subspace of the Nakano space l(pn ) if and only if p ∈ Lim(pn ).

•

(Hudzik-Wu-Ye 1994): The Kottman packing constant Λ(l(pn ) ) of the Nakano6 space l(pn ) is equal to 1/2 if lim supn→∞ pn = ∞ and 21/p /(2 + 21/p ) if lim supn→∞ pn < ∞, where p = lim inf n→∞ pn .

• (Cui-Hudzik 1998): The Nakano space l(pn ) has the uniform Opial property7 if and only if the lim supn→∞ pn < ∞. •

5

(Casazza-Kalton 1998): Let 1 < pn < ∞ and pn ↓ 1. Suppose 1 that for some constant a > 0 we have p2n − p1n ≤ lnan for n ≥ 2. Then the Nakano space l(pn ) has a unique unconditional basis.8

A Banach space X is said to have the Dunford-Pettis property if xn → 0 weakly in X and fn → 0 weakly in the dual space X ? imply that fn (xn ) → 0. 6 The Kottman packing constant Λ(X) of a Banach space X is a number defined by Λ(X) = sup{r > 0 : there exists a sequence {xn }∞ n=1 ⊂ BX , kxm −xn k ≥ 2r for m 6= n and ∪∞ n=1 Bx (xn , r) ⊂ BX }, where BX denotes the unit ball of X and BX (x, r) denotes the ball in X centred at x ∈ X and with radius r > 0. Kottman in 1970 introduced a parameter D(X) = sup inf m6=n kxm − xn k, where the supremum is taken over all sequences (xn ) from the unit sphere of X and proved that Λ(X) = D(X)/(2+D(X)), 7 A Banach space X is said to have the uniform Opial property if for every ε > 0 there exists τ > 0 such that for every weakly null sequence {xn } in the unit sphere of X and x ∈ X with kxk ≥ ε it holds that 1 + τ ≤ lim inf n→∞ kxn + xk. 8 A Banach space X with an unconditional basis has a unique unconditional basis if any two normalized unconditional bases are equivalent after a permutation, that is, if (un ) and (vn ) are two unconditional basic sequences and there is a permutation π on N so that (un ) and (vπ(n) ) are equivalent. It is well-known that l2 has a unique unconditional basis and a classic result of Lindenstrauss and PeÃlczy´ nski (1968) asserts

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We should mention here that Sundaresan use in his proof the general theorem of Nakano ([Nb10], Thm 3, pp. 227–228) on modulars to the concrete modular ∞ X 1 m(x) = |xn |pn . pn n=1

THEOREM 2 (Nakano, 1951). If a modular m is uniformly convex (∀r, ε > 0 ∃δ > 0 m(x) ≤ r, m(y) ≤ r, m(x − y) ≥ ε =⇒ m( x+y 2 ) ≤ 1 1 m(x)+ m(y)−δ), uniformly finite (sup m(λx) < ∞ ∀λ > 0) and m(x)≤1 2 2 uniformly simple (inf m(x)≥1 m(λx) > 0 ∀λ > 0), then the norm induced by the modular m (kxk = inf{ε > 0 : m(x/ε) ≤ 1}) is uniformly convex. The result of Nakano on the Schur property of Nakano sequence space l{pn } was generalized to variable Orlicz spaces l{ϕn } by Yamamuro [Ya54], Wnuk [Wn91], [Wn93], Kami´ nska-MastyÃlo [KM02] and Zlatanov [Zl07]. For a measurable function p : I → [0, ∞], where I = [0, 1] (Nakano case) or in general I = Ω we denote by Lp(t) the Nakano function spaces called also the variable Lebesgue spaces or variable Lp spaces by Lp(t) = {x ∈ L0 (I) : m(x/λ) < ∞ for some λ > 0}, where

Z |x(t)|p(t) dt (Orlicz, 1931)

m(x) = I

or

Z m(x) = I

1 |x(t)|p(t) dt (Nakano, 1951) p(t)

1 ∞ with the convention ∞ α = 0 for 0 ≤ α ≤ 1 and = ∞ for α > 1. These spaces are Banach spaces with the Luxemburg-Nakano norm

kxkLp(t) = inf{λ > 0 : m(x/λ) ≤ 1} and the Amemiya-Orlicz norm kxkA = inf Lp(t)

k>0

1 + m(kx) . k

that the spaces l1 and c0 also have unique unconditional bases.

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L. Maligranda p(t)

Denote La = {x ∈ L0 (I) : m(x/λ) < ∞ for all λ > 0}. Nakano proved the following result concerning the duality and uniform convexity of these spaces (cf. [Nb10], pp. 236–240): THEOREM 3 (Nakano, 1951) (a) If p is bounded on I, then the p(t) 1 1 dual space (La )? = Lq(t) , where p(t) + q(t) = 1 for almost all t ∈ I. (b) Space Lp(t) is regular9 if and only if 1 < ess inft∈I p(t) ≤ esssupt∈I p(t) < ∞. (c) If 1 < inft∈I p(t) ≤ supt∈I p(t) < ∞, then Lp(t) is uniformly convex and uniformly smooth in both norms (Luxemburg-Nakano and Amemiya-Orlicz). Let us mention that the investigations of operators between Nakano function spaces (variable Lebesgue spaces) are just now very popular. Mathematicians are proving boundedness of maximal operator and some other classical operators in Lp(t) spaces (L. Diening, D. Cruz-Uribe, D. E. Edmunds, X. Fan, A. Fiorenza, P. H¨ast¨ o, A. Yu. Karlovich, V. Kokilashvili, T. Kopaliani, A. K. Lerner, A. Nekvinda, C. P´erez, L. Pick, M. Ruˇziˇcka, N. Samko, S. Samko). A good information source is the survey article of Samko [Sa05] with its references given there. We should mention that in the proofs new techniques are necessary since these spaces are NOT symmetric (rearrangement-invariant), that is, two equimeasurable functions can have different norms. For example, take on [0, 1] with the Lebesgue measure the function p(t) = 2 on [0, 1/2] and 3 on (1/2, 1]. Then the functions x = χ[0,1/2] and y = χ[1/2,1] are equimeasurable, but have different norms kxk = 2−1/2 and kyk = 2−1/3 . In general, the space Lp(t) is rearrangement invariant if and only if p(t) is equivalent to a constant function a.e. on I. Some more papers with the word Nakano sequence or function spaces in the title are: Waterman-Ito-Barber-Ratti [WIBR69], Wnuk [Wn91], Suantai [Su99], Dhompongsa [Dh00], Blasco-Gregori [BG01], Karlovich [Ka07], Poitevin-Raynaud [08] and Yaacov [Ya09]. 9

A modular m or modular space X is said to be regular if the space X and ¯ coincide, or equivalently, if m = m, ¯ where m the second modular adjoint X ¯ = ¯ where X ¯ = {ϕ : X → R : sup supx∈X [¯ x(x) − m(x)] for x ¯ ∈ X, m(x)≤1 |ϕ(x)| < ∞} – modular bounded functionals.

Hidegoro Nakano

123 3b. Orlicz-Nakano spaces

Let (Ω, µ) be a σ-finite measure space and M : [0, ∞] × Ω → [0, ∞] be a non-decreasing convex function in first variable with M (0, t) = 0 and measurable in the second variable. The space LM = {f ∈ L0 (Ω) : kf kM = inf{λ > 0 : IM (f /λ) ≤ 1} < ∞, R with the modular IM (f ) := Ω M (|f (t)|, t)dµ(t) was introduced by Nakano in 1950 as a generalization of the Orlicz space [i.e. when the function M depends only on the first variable M (u, t) = M (u)] and had (and should have) the name Nakano space or Orlicz-Nakano space. Sometimes this space is called Orlicz space with a parameter. Nakano himself used the name modulared function space (cf. [Nb8], appendix). The name Orlicz-Nakano space for this space was used in the title of the papers, for example, by the following authors: Shragin [Sh67], [Sh75] and Levchenko-Shragin [LS68]. In 1959 J. Musielak and W. Orlicz considered these spaces without the assumption of convexity on the function M of two-variables and as special case they have also the convex case. Now, most mathematicians use the name Musielak-Orlicz spaces for these spaces (unfortunately even in the convex case, i.e., the case considered by Nakano already in 1950). The special case of these spaces when M (u, t) = up(t) has the name Nakano spaces (1951) but they were investigated (both in the sequence and function case) already in 1931 by Orlicz. I like to say in such situation (see L. Maligranda, Why H¨older’s inequality should be called Rogers’ inequality, [Ma98], page 80): Some mathematicians are lucky and some other unlucky in getting their names to the results they proved. Orlicz spaces with a parameter were and still are investigated by mathematicians from all of the world. Let us mention some important places and/or names: Sapporo (Nakano school: T. Ando, J. Ishii, T, Ito, T. Shimogaki, S. Yamamuro), Russia (I. V. Shragin, A. I. Skorik), Pozna´ n (Orlicz and Musielak school with important results of L. Drewnowski, H. Hudzik, A. Kami´ nska and W. Wnuk), China (Harbin school: S. Chen, Y. Cui, Z. Shi, T. Wang), France (Y. Raynaud, P. Turpin), Israel (J. Lindenstrauss, L. Tzafriri, J. Y. T. Woo), Spain (O. Blasco, F.

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L. Hern´andez, V. Peirats, B. Rodriguez-Salinas and C. Ruiz), Thailand (S. Dhompongsa, S. Saejung, S. Suantai), Bulgaria (R. Maleev, B. Zlatanov), Germany (M. Denker, L. Diening, T. Landes) and USA (J. E. Jamison, A. Kami´ nska, P.-K. Lin). 3c. Modular vector lattices and modular spaces On a Dedekind complete vector lattice X (Nakano used term universally continuous semi-ordered linear space) Nakano introduced in 1947 (published in 1950 Modulared Semi-Ordered Linear Spaces [Nb8]) a modular m, that is, a function m : X → [0, ∞] such that (1) if m(ξx) = 0 for all ξ ≥ 0, then x = 0, (2) to any x ∈ X there exists an α > 0 such that m(αx) < ∞, (3) m(ξx) is a convex function of ξ ≥ 0, (4) if |a| ≤ |b|, then m(a) ≤ m(b), (5) if x ∧ y = 0, then m(x + y) = m(x) + m(y), (6) if 0 ≤ aλ % a, then m(a) = sup m(aλ ). A Dedekind complete vector lattice X with a modular on it is called a modular space or the Nakano space as, for example, in the paper by Yamamuro [Ya59]. In the preface of the book Modulared Semi-Ordered Linear Spaces [Nb8] (1950) on page 2 we can find the following information of Nakano (Tokyo, March 1950): 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 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 relay upon future research, but I can believe from fundamental properties that modulared semi-ordered linear spaces will play an important part in the future mathematics.

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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 came 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. By the way in the same book [Nb8] on page 7 there is a reference: H. Nakano, Modulared linear spaces (not published). Remark 1. In Nakano’s paper [N51a], which was sent to the journal on 16 June 1948 it was mentioned that the conception of modulars appeared in earlier papers, which were published in Japanese language in the journal Functional Analysis (in years 1947–1948): [N1], [N2], [N4] and in the paper [N51b] Nakano informed on page 512 about his paper in Japanese [N15]. Immediate question: what was the journal Functional Analysis in Japan? Was it a journal or just preprints? Can we anywhere find copies of it? Professor T. Ando informed me at the conference that this was in fact a journal and that he has some copies. Later on I have even got from him xerox-copies of contents of the journal. Journal was published at the Department of Mathematics, Faculty of Science, University of Tokyo. It started with Volume I, no. 1 on 25 December 1947 and ended at Volume III, no. 1, on 20 October 1954 and in his opinion the name Functional Analysis appeared here for the first time in Japan. Really, K. Yosida published the original Japanese version of his famous Springer monograph Functional Analysis (1965) in 1951 with Japanese title Isou Kaiseki (Topological Analysis). The following classical result appeared in the Nakano book from 1950 on modulared spaces [Nb8]: THEOREM 4 (Nakano, 1950). In the modular space X we have two norms 1 + m(λx) 1 : m(λx) ≤ 1} and kxk2 = inf , kxk1 = inf{ λ>0 |λ| λ

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which are equivalent kxk1 ≤ kxk2 ≤ 2 kxk1 and kxk1 ≤ 1 if and only if m(x) ≤ 1. Both norms have the Fatou property in X 10 , and, therefore X is a Banach space with each norm. The first norm is usually called the Luxemburg norm, but I am using from 1989 the proper name the Luxemburg-Nakano norm since Nakano introduced it in 1950 and Luxemburg in 1955 in his PhD thesis [Lu50], and only for the Orlicz spaces (Luxemburg has Nakano both books from 1950 and 1951 in the references [Lu50], p. 43). The second norm is called the Amemiya norm (or the Amemiya-Orlicz norm). H. Nakano in his review MR0106412 (21#5144) of the Krasnosel’ski˘ıRuticki˘ı book [KR58] on Orlicz spaces wrote the following: Here the authors quote the book: H. Nakano, Modulared semi-ordered linear spaces, [Maruzen, Tokyo 1950; MR38565 (12, 420a)], and say that the modulared semi-ordered linear spaces are related to Orlicz spaces. However, they seem not to be acquainted with the fact that Orlicz spaces are completely contained in modulared semi-ordered linear spaces as special examples. Most of the facts in this chapter about Orlicz spaces have already been obtained as properties of modulared semi-ordered linear spaces. Especially, the norm called by the name Luxemburg, related to W. A. J. Luxemburg’s thesis [Technische Hogeschool te Delft, 1955; MR0072440 (17, 285a)], is defined as the second norm in Nakano’s book.

Proof of Theorem 4. Observe that we can write kxk1 = inf{λ > 0 : m(x/λ) ≤ 1}. We only show the triangle inequality for both norms to see how the convexity property of the modular m is working in both cases. Let 10

A norm (X, k · k) has the Fatou property in X if 0 ≤ xn % x and supn kxn k < ∞, then x ∈ X and kxn k % kxk.

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127

a = kxk1 > 0 and b = kyk1 > 0. Then, by the convexity of modular m, µ m

x+y a+b

¶

µ

¶ a x b y = m · + · a+b a a+b b a x b y ≤ m( ) + m( ) a+b a a+b b a b ≤ ·1+ · 1 = 1, a+b a+b

and so kx + yk1 ≤ a + b = kxk1 + kyk1 . For the second norm, for arbitrary ε > 0 we can find λ > 0, µ > 0 such that 1 + m(λx) 1 + m(µy) < kxk2 + ε, < kyk2 + ε. λ µ Then, by the convexity of m, kx + yk2 ≤ = ≤ =

λµ 1 + m( λ+µ (x + y)) λµ λ+µ

· ¸ λ+µ µ λ 1 + m( λx + µy) λµ λ+µ λ+µ · ¸ µ λ λ+µ 1+ m(λx) + m(µy) λµ λ+µ λ+µ 1 1 m(λx) m(µy) + + + < kxk2 + kyk2 + 2ε, µ λ λ µ

and the proof is complete. Remark 2. Modulars in the Nakano sense give only the Orlicz spaces with a parameter. In fact, L. Drewnowski and W. Orlicz [DO68] proved in 1968 that if X is a sublattice in L0 (Ω) containing with each function x also all functions of the form xχA , then an orthogonally additive modular m in the sense of Nakano has a representation of the form Z m(x) = M (|x(t)|, t) dµ(t), (1) Ω

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where M : [0, ∞) × Ω0 → [0, ∞] satisfies all conditions imposed on a Musielak-Orlicz function except the continuity of M (·, t) at the point u = 0; Ω0 is the support of the space X. In 1959 J. Musielak and W. Orlicz [MO59] defined modular % on the real vector space X (modular in the sense of Musielak-Orlicz) as the function % : X → [0, ∞] satisfying: (%1)

%(x) = 0 ⇔ x = 0,

(%2)

%(−x) = %(x),

(%3)

%(αx + βy) ≤ %(x) + %(y), for

α, β ≥ 0, α + β = 1,

and the modular space X% = {x ∈ X : limt→0+ %(tx) = 0}. As they showed on X% we can define the Mazur-Orlicz F -norm by µ ¶ x ≤ λ}, |x|% = inf{λ > 0 : % λ and |xn |% → 0 if and only if %(txn ) → 0 for all t > 0. If the modular % is, in addition, convex, that is, we have not only property (%3) but also %(αx + βy) ≤ α%(x) + β%(y) for α, β > 0, α + β = 1,

(2)

then on X% we can define two norms in the same way as for the modular in the sense of Nakano: kxk1 = inf{λ > 0 : %(x/λ) ≤ 1},

1 + %(λx) . λ>0 λ

kxk2 = inf

(3)

Moreover, as it was proved in 1961 by Orlicz [Or61] we have the following equality: 1 %(λx) }. (4) kxk1 = inf max { , λ>0 λ λ Proof of (4). If %(λx) ≤ 1, then kxk1 ≤ the convexity of % we obtain that %(

x λ−1 %(λx)

) = %(

1 λ.

If %(λx) > 1, then by

λx %(λx) )≤ = 1, %(λx) %(λx)

whence kxk1 ≤ %(λx)/λ and so 1 kxk1 ≤ max { , %(λx)/λ}. λ

Hidegoro Nakano

129

On the other hand, if kxk1
0 λ λ λ0 λ0 λ0 Since 1/λ0 can be arbitrary close to kxk1 , we get that 1 %(λx) inf max { , } ≤ kxk1 , λ λ

λ>0

and the proof is complete. In the case of Orlicz spaces Nakano (1951) and Luxemburg-Zaanen (1956) proved that the second norm is the same as the Orlicz norm (under additional assumptions that the Orlicz function M is finitevalued and limu→∞ Mu(u) = ∞; in general this was proved by HudzikMaligranda in 2000): ¯ ½¯Z ¾ ¯ ¯ ∗ 0 kxk2 = kxkM := sup ¯¯ x(t)y(t) dµ¯¯ : y ∈ LM , %M ∗ (y) ≤ 1 , (5) R

Ω

M ∗ (|y(t)|) dµ

where %M ∗ (y) = Ω and the complementary function M ∗ : [0, ∞) → [0, ∞] to M is defined by M ∗ (u) = sup{uv − M (v) : v ≥ 0}. Note that equality (5) for Nakano-Orlicz spaces (Orlicz spaces with parameter) was proved recently by Fan [Fa07]. We should however mention that the last result was already proved in 1967 by Shragin in his overlooked paper [Sh67]. This paper is very difficult (almost impossible) to find in libraries, but I have got its copy personally from Shragin in 2009. %

Remark 3. Nakano didn’t define the modular convergence: xn −→ 0 if %(λxn ) → 0 for some λ > 0. This convergence is not in general topological in the sense that we cannot define topology τ on X% in such a way % τ that xn −→ 0 ⇔ xn −→ 0. Nakano (1950) is calling the convergence %(λxn ) → 0 for all λ > 0 as modular convergence but this is just norm convergence kxn k% → 0. We finish this part by mention some authors of monographs containing the theory of modular spaces: Nakano (1950), Nakano (1951), Musielak (1978), Musielak (1983), Wnuk (1984) and Maligranda (1989).

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L. Maligranda 3d. Vector lattices and locally solid Riesz spaces

H. Nakano belongs to a group of mathematicians who created the foundations of the theory of Riesz spaces (= vector lattices) and topological Riesz spaces (= locally solid Riesz spaces). His fundamental contribution concerns all basic aspects of the theory are in: (a) algebraic and order structure of Riesz spaces, (b) relationships joining the order and topological structures of topological Riesz spaces, (c) order bounded linear functionals and operators. A. C. Zaanen in his review of the book by H. H. Schaefer [Sc74] Banach Lattices and Positive Operators [MR0423039 (54 #11023)] did the following comments about Nakano’s research and notations: (...) between 1940 and 1944, several Japanese mathematicians made important contributions to the subject (H. Nakano, T. Ogasawara, K. Yosida). (...) In the same year, 1950, H. Nakano’s book [Modulared semiordered linear spaces, Maruzen, Tokyo 1950 ] came out, but this work was of somewhat limited value because of the unusual terminology, the absence of examples and the restriction to the author’s own research. In what follows the vector lattice (= Riesz space) E is a real vector space with order relation ≤ on E compatible with the algebraic structure and for every pair of vectors x, y ∈ E the supremum x ∨ y = sup{x, y} and infimum x ∧ y = inf{x, y} exist. The concept of a Riesz space is due independently to F. Riesz (1928), H. Freudenthal (1936) and L. V. Kantorovich (1937). 3d1. Order completeness of C(K) We start with order completeness of vector lattices, which has similar fundamental role in vector lattices as topological completeness in normed spaces. A vector lattice (=Riesz space) E is order complete [σ-order complete] whenever order bounded from above subsets [countable subsets] of E have the supremum (least upper bound) or equivalently: order

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131

bounded from below subsets [countable subsets] of E have the infimum (greatest lower bound). Thus, E is order complete ⇐⇒ every net 0 ≤ xα %≤ x has a supremum ⇐⇒ every net 0 ≤ xα & has an infimum. • Of course, Rn is order complete and the space C[0, 1] fails to be order complete (is even not σ-order complete) since for  if 0 ≤ t ≤ 21 ,  0, 1 linear and continuous, if 12 ≤ t ≤ 2+n , fn (t) =  1 1, if 2+n ≤ t ≤ 1, we have that 0 ≤ fn %≤ 1, and fn → f0 pointwise with f0 ∈ / C[0, 1]. • For a measure space (Ω, Σ, µ) the space L0 (Ω, µ) of all µequivalence classes of real-valued Σ-measurable functions on Ω is σ-order complete vector lattice with natural order f ≤ g if f (t) ≤ g(t) for µ-almost every t ∈ Ω. • Any ideal E ⊂ L0 (Ω, µ) with µ σ-finite (that is, if g ∈ E, f ∈ L0 and |f | ≤ |g| implies f ∈ E) is order complete. Order complete [σ-order complete] vector lattices are also called Dedekind complete [σ-Dedekind complete] lattices (Luxemburg-Zaanen, 1977) or conditionally complete (Vulich, 1966). It is natural to ask when such classical spaces as the space C(K) of continuous real functions on a topological space K is order complete (Dedekind complete) with the standard pointwise order, i.e., f ≤ g if and only if f (t) 6 g(t) for every t ∈ K. The answer was given independently by Nakano [N41f] and Stone [St49]. THEOREM 5 (Nakano, 1941; Stone, 1949). For a topological space K we have: (a) If K is extremally disconnected (i.e., the closure of every open set is open), then C(K) is Dedekind complete. (b) If C(K) is Dedekind complete and K is also completely regular, then K is extremally disconnected. The following corollary is an immediate consequence of the previous result:

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Corollary 1. A compact topological space K is extremally disconnected if and only if C(K) is Dedekind complete (order complete Banach lattice). Theorem 5 appeared as the Nakano-Stone theorem, for example, in the following classical books (ordered by year): Peressini [Pe67], Proposition in part (1.7) on page 16; Luxemburg-Zaanen [LZ71], Thm 43.9, p. 287 and Thm 43.11, p. 288; Diestel-Uhl [DU77], Lemma 8, pp. 154–155; Lindenstrauss-Tzafriri [LT79], Proposition 1.a.4, pp. 4–5; and Aliprantis-Burkinshaw [AB03], Thm 1.50, p. 24 and Corollary 1.51, p. 25. 3d2. Order completion of vector lattices Various constructions, called enlargements, are known in mathematics. They allow us to embed spaces into other spaces possessing better properties (for instance in complete spaces – such enlargements are called completions, or in compact spaces which are said to be compactifications etc.). An embedding, in this context, means an operator preserving the structure of a space, for example a homomorphism in the case of topological spaces. Moreover, an enlargement should not be too large – usually it is required that an image of the initial space is dense in an enlargement. An enlargement procedure exists also in the theory of Riesz spaces. The most famous result in this direction was obtained independently by Nakano [N41c] and Judin [Ju41], and it says that every Archimedean Riesz space E (i.e., inf n1 x = 0 for every positive x ∈ E) embeds into its Dedekind completion E δ with preservation of suprema and infima. The exact form of this theorem reads as follows: THEOREM 6 (Nakano, 1941; Judin, 1941). Let E be an Archimedean Riesz space. Then there exists a Dedekind complete Riesz space E δ (unique up to Riesz isomorphism) such that E δ contains a majorizing order dense Riesz subspace that is order isomorphic to E, which we identify with E. That is, E is identified with a Riesz subspace of E δ and for each x ∈ E δ we have x = sup{y ∈ E : y 6 x} = inf{z ∈ E : z > x}. Note that T : E → F is a Riesz isomorphism if T (x ∨ y) = T x ∨ T y

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133

for all x, y ∈ E. Theorem 6 appeared as the Judin-Nakano theorem, for example, in the following classical books: Luxemburg-Zaanen [LZ71], Thm 32.5, p. 191; Aliprantis-Burkinshaw [AB78], Thm 2.3, p. 10; and AliprantisBurkinshaw [AB03], Thm 1.41, p. 19. 3d3. Representation of abstract Lp lattices A normed Riesz space is a Riesz space equipped with a lattice norm (|x| ≤ |y| =⇒ kxk ≤ kyk). A Banach lattice is a complete normed Riesz space. A Banach lattice (E, k · k) is said to be an abstract Lp -space for some p ∈ [1, ∞), whenever its norm is p-additive in the sense that kx + ykp = kxkp + kykp holds for all x, y ∈ E with x ∧ y = 0. The following famous results asserts that every abstract Lp -space has a representation of the form Lp (µ). For p = 1 it was proven by S. Kakutani [Ka41] while H. F. Bohnenblust [Bo40] and H. Nakano [N41e] (c.f. also [Nb13, p. 121]) established the general result under some additional conditions. THEOREM 7 (Kakutani, Bohnenblust, Nakano, 1941). Any abstract Lp -space E is lattice isometric to a concrete Banach lattice Lp (µ), where µ is the direct sum of finite measures. If E has a weak unit, then µ can be chosen to be a finite measure. Note that an element e ≥ 0 of a Banach lattice E is said to be a weak unit of E if e ∧ x = 0 for x ∈ E implies that x = 0. Theorem 7 appeared as the Kakutani-Bohnenblust-Nakano theorem or Kakutani-Nakano theorem or Bohnenblust-Nakano theorem, for example, in the following classical books: Semadeni [Se71], Notes: Nakano (1941/42), p. 232; Lacey [La74], Thm 3, p. 135 (BohnenblustNakano theorem); Aliprantis-Burkinshaw [AB78], Thm 10.12, pp. 71–73; Lindenstrauss-Tzafriri [LT79], Proposition 1.b.2, p. 15; Niculescu-Popa [NP81], Thm 2.6.1, p. 77 (Kakutani-Nakano theorem); Schwarz [Sc84], Thm 11.4, p. 120–121; Aliprantis-Burkinshaw [AB85], Thm 12.26, pp. 192–193 and Abramovich-Aliprantis [AA02], Thm 3.5, p. 95.

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L. Maligranda 3d4. Nakano’s results on order continuous functionals

The most important class of linear operators acting between Riesz spaces E, F consists of order bounded operators, i.e., linear maps T : E → F transferring order bounded sets into order bounded sets. An order bounded set is the set contained in some order interval. The class of order bounded operators is denoted by Lb (E, F ). If F is Dedekind complete, then Lb (E, F ) forms a Dedekind complete Riesz space with respect to the natural pointwise order: S ≤ T if and only if S(x) ≤ T (x) for every x ∈ E+ . The particular case F = R is a matter of special importance. The space Lb (E, R) is called the order dual of E and is denoted by E ∼ . Note that the class of regular operators Lr (E, F ), that is, those which can be written as a difference of two positive operators is a proper vector subspace of Lb (E, F ), i.e., Lr (E, F ) ⊂ Lb (E, F ) ⊂ L(E, F ) and both inclusions may be proper. Nakano introduced in 1950 in his book [Nb8] two basic notions which persistently entered to the terminology of Riesz spaces: En∼ of order continuous and Ec∼ of σ-order continuous functionals. The order continuity (o)

(σ-order continuity) of f means that xα → x implies f (xα ) → f (x) (o)

(xn → x =⇒ f (xn ) → f (x)), i.e., f preserves order convergence of nets (sequences, respectively). Note that a net {xα } of a Riesz space E order converges to x in E, (o)

denoted by xα → x, if there exists a net {yα } of E such that |xα − x| ≤ yα ↓ 0 holds in E. Similarly also for sequences. Clearly, En∼ ⊂ Ec∼ ⊂ E ∼ and E ∗ ⊂ E ∼ , and all inclusions may be proper. For example, if E = Lp (µ), 1 ≤ p < 0 ∞, then E ∗ = E ∼ = Ec∼ = En∼ = Lp (µ) and if E = C[0, 1], then En∼ = Ec∼ = {0} and E ∗ = E ∼ is the Riesz space of all regular Borel measures on [0, 1]. If E is a Banach function space (in the sense of Luxemburg-Zaanen) over a σ-finite measure, then the classes En∼ , Ec∼ of order continuous and

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σ-order continuous functionals coincide and this common class is known as the K¨othe dual of E. RElements of the K¨othe dual have a nice integral representation: f (x) = Ω xy dµ where y is a fixed measurable function determined by f and x runs over E. Nakano characterized in 1950 disjointness of order continuous functionals by inclusions of their carriers and null ideals known as Nakano’s carrier theorem ([Nb8], Thm 20.1, p. 74). For f ∈ En∼ consider the null ideal of f defined by Nf = {x ∈ E : |f |(|x|) = 0} and the carrier of f Cf = {x ∈ E : |x| ∧ |y| = 0 for all y ∈ Nf }. THEOREM 8 (Nakano, 1950). Let E be σ-Dedekind complete and the functionals f, g ∈ En∼ . Then the following statements are equivalent: (a)

|f | ∧ |g| = 0.

(b)

Cf ⊂ Ng .

(c)

Cg ⊂ Nf .

(d)

Cf ∧ Cg = 0, that is, |x| ∧ |y| = 0 for all x ∈ Cf , y ∈ Cg .

It is worth to add that later on Luxemburg and Zaanen showed that the above theorem remains valid for all Archimedean Riesz spaces. Theorem 8 appeared as the Nakano theorem in the following classical books: Aliprantis-Burkinshaw [AB78], Thm 3.10, p. 24; [AB85], Thm 5.2, p. 56; Meyer-Nieberg [Me91], Thm 1.4.11, p. 37; AbramovichAliprantis [AA02], Ex. 22, p. 27; Abramovich-Aliprantis [AA02a], Problem 1.2.22, p. 31; and Aliprantis-Burkinshaw [AB03], Thm 1.82, p. 42. The evaluation κ : E → E ∼∼ (= (E ∼ )∼ ) given by the formula (κ(x))(f ) = f (x) defines an embedding taking values in (E ∼ )∼ n . If I ⊂ E ∼ is an ideal, then κ restricted to I defines a natural embedding of E into In∼ . Nakano investigated in the book [Nb8] (Thm 22.6, p. 83) the properties of this embedding (see also Aliprantis-Burkinshaw [AB03], Thm 1.83, p. 43): Let E be a Riesz space, let I be an ideal of E ∼ , and let κ be the natural embedding of E to In∼ . Then we have:

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(a) κ is a Riesz homomorphism, i.e., |κ(x)| = κ(|x|). In particular κ(E) is a Riesz subspace of In∼ . (b) κ is an order isomorphism (into) if and only if I separates points of E. (c) If E is Archimedean and I ⊂ En∼ , then κ is order continuous and κ(E) is order dense in In∼ , i.e., for every 0 < F ∈ In∼ there exists x ∈ E satisfying 0 < κ(x) 6 F . The theorem mentioned above is the main tool in investigations of so called perfect Riesz spaces E, i.e., such spaces that κ(E) = (En∼ )∼ n. Nakano is the author of several basic results concerning perfect spaces (they were published in 1950 in his book [Nb8], Section 24; see also Aliprantis-Burkinshaw [AB85], Thm 1.71, p. 63; and Aliprantis-Burkinshaw [AB03], Thms 3.17–3.19, pp. 83–84). A Riesz space E is a perfect Riesz space if and only if the following two conditions holds: (a) En∼ separates the points of E. (b) Whenever a net (xα ) in E satisfies 0 6 xα ↑ and supα f (xα ) < ∞ for each 0 6 f ∈ En∼ , then there exists some x ∈ E satisfying xα ↑ x (x is the supremum, in E of the net xα ). Applying the above characterization of perfect Riesz spaces Nakano showed the following: E ∼ is perfect for every Riesz space E and that every band (= order closed ideal) of a perfect Riesz space is a perfect Riesz space in its own right. Nakano used the name “normal manifold” instead of “band”. Nakano [Nb8], Thm 31.1, p. 134 (see also Aliprantis-Burkinshaw [AB03]–the text before Thm 2.22, p. 57) was the first who noticed that the topological dual E ∗ is an ideal of order dual E ∼ for an arbitrary normed Riesz space E (i.e., continuous linear functionals form an ideal in the order dual). This fundamental result remains valid for an arbitrary topological Riesz space. On the other hand E ∗ = E ∼ for a Banach lattice E.

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3d5. Selected Nakano’s results on topological Riesz spaces When we shall present Nakano’s contributions to the theory of Riesz spaces theory it is impossible to omit his achievements in investigations of Levi and Fatou topologies. The topological Riesz space means a pair (E, τ ), where E is a Riesz space and τ is a locally solid linear topology on E, i.e., τ possesses a basis of neighborhoods at zero consisting of solid sets. If τ is metrizable, then there exists a monotone F-norm k · k generating τ : |x| 6 |y| ⇒ kxk 6 kyk. It is well known that decreasing norm convergent nets in topological Riesz spaces are order convergent to the same limit. If the converse τ holds, i.e., if xα ↓ x implies xα → x, then we say that the topology τ is order continuous or that τ is a Lebesgue topology. Topological Riesz space (E, τ ) satisfies the Lebesgue property (or τ τ is a Lebesgue topology) if xα ↓ 0 in E implies that xα → 0. Replacing net by sequence we will have the definition of the σ-Lebesgue property. The σ-Lebesgue property first appeared in the works of Kantorovich and its systematic studies for normed Riesz spaces were done by Nakano under the name universally continuous and continuous norm in the case of sequences (see Nakano [Nb8], pp. 126–134; see also AliprantisBurkinshaw [AB06], p. 75). A few elementary characterizations of Banach lattices with order continuous norms are included in the next theorem. They are essentially due to Nakano [Nb13], p. 321(see also Aliprantis-Burkinshaw [AB06], Theorem 4.9, p. 186 and Lindenstrauss-Tzafriri [LT79], Thm 1.b.16, p. 27): equivalence (a) ⇐⇒ (d) and p. 28: equivalence (a) ⇐⇒ (e)): THEOREM 9 (Nakano, 1955). For a Banach lattice E the following statements are equivalent. (a) E has order continuous norm. (b) If 0 6 xn ↑6 x holds in E, then (xn ) is a norm Cauchy sequence. (c) E is Dedekind complete and xn ↓ 0 in E implies that kxn k ↓ 0. (d) E is an ideal in E ∗∗ . (e) Each order interval in E is weakly compact.

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A locally solid Riesz space (E, τ ) is said to satisfy the Levi property or that τ is a Levi topology if every increasing τ -bounded net of positive elements has a supremum in E. By replacing the word net with sequence in the preceding definition we obtain the definition of the σ-Levi property or a σ-Levi topology. Clearly, in the case of normed Riesz spaces the Levi property (=weak Fatou property in the sense of Luxemburg-Zaanen 1983) can be expressed as follows: if 0 6 xα ↑ and sup kxα k < ∞, then sup xα exists in E. α

α

Note that metrizable σ-Levi topologies are complete. Levi topologies were first considered by Nakano in 1953 in the paper [N53] (see also Aliprantis-Burkinshaw [AB03], footnote on p. 83). The main examples of spaces with the Levi property are: (a) dual Banach lattices, (b) C(K) with K extremally disconnected, (c) L0 (µ) – the space of all (equivalence classes of) measurable functions with µ σ-finite, where the topology τ (µ) of convergence in the µ-measure on sets of finite measure is a Lebesgue and Levi topology, (d) if (E, τ ) is a topological Riesz space such that E is a Riesz subspace of L0 (µ) and the natural embedding E ,→ L0 (µ) is continuous, then τ is a Levi topology whenever τ possesses a basis of τ (µ)closed neighborhoods of zero. In particular, standard topologies in Lp -spaces (0 < p 6 ∞), Orlicz spaces, Lorentz spaces are Levi. On the other hand, the norm topology in c0 is a standard example of a topology which is not σ-Levi. Moreover, l∞ /c0 is not σ-Levi and lc∞ (Γ) is σ-Levi but not Levi. Using the notion of Levi topology Nakano characterized perfect spaces in the following way: A Riesz space E is perfect if and only if the absolute weak topology |σ|(E, En∼ ) is a Levi topology. Banach lattices satisfying both Lebesgue and Levi properties are of special importance and they are called KB-spaces. KB-spaces coincide with the class of spaces which do not contain any closed linear subspace isomorphic (in the sense of Banach spaces theory) to c0 .

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A locally solid Riesz space (E, τ ) is said to satisfy the Fatou property (or that τ is a Fatou topology) if τ has a base at zero consisting of solid and order closed sets. Similarly, (E, τ ) satisfies the σ-Fatou property (or that τ is a σ-Fatou topology) if τ has a base at zero consisting of solid and σ-order closed sets. It is well known that every Lebesgue topology is a Fatou topology. Moreover, τ is Fatou (σ-Fatou) if and only if there exists a family of Fatou (σ-Fatou) F-seminorms that generates the topology τ . Let us recall that an F-seminorm k·k is said to be a Fatou F-seminorm (σ-Fatou F-seminorm) if 0 6 xα ↑ x implies kxα k ↑ kxk (0 6 xn ↑ x ⇒ kxn k ↑ kxk). Fatou and σ-Fatou norms were first studied by Nakano in 1950 in his book [Nb18, §30] under a different terminology (see also AliprantisBurkinshaw [AB03], p. 100, historical note). The Fatou property is closely related to topological completeness. The result below due to Nakano [N53, Thms 3.3 and 4.2] is regarded as one of the deepest results in the theory of topological Riesz spaces. THEOREM 10 (Nakano, 1953). If (E, τ ) is a Dedekind complete topological Riesz space with the Fatou property, then the order intervals of E are τ -complete Nakano’s original proof, however, has a gap in a nonmetrizable case that was filled in later by H. H. Schaefer [Sc60, Thm 1, pp. 304– 305]). Another proof of Nakano theorem 10 was found by AliprantisBurkinshaw in [AB75] and [AB76]. Theorem 10 appeared as the Nakano theorem in the following classical books: Peressini [Pe67], Thm 1.3, p. 140; Aliprantis-Burkinshaw [AB78], Thm 13.1, p. 90; and Aliprantis-Burkinshaw [AB03], Thm 13.1, p. 90. Kantorovich and Akilov quote in [KA77, Thm 7, p. 388] the following characterization of the Fatou property which is due to Mori, Amemiya and Nakano [N55] (see also Meyer-Nieberg [Me91], Thm 2.4.21, p. 98): For a normed Dedekind complete Riesz space (E, k · k) such that En∼ separates points the following statements are equivalent: (a) The norm k · k is a Fatou norm.

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(b) kxk = sup{|f (x)| : f ∈ E ∗ ∩ En∼ , kf k 6 1} for every x ∈ E. (c) The canonical embedding κ : E → (En∼ ∩ E ∗ )∗ preserves norms. Another result by Nakano concerning topological completeness was proved in [N53, Thm 4.2, p. 96] (see also Aliprantis-Burkinshaw [AB03], Thm 4.37, p. 112): THEOREM 11 (Nakano, 1953). If a locally solid topology is both Levi and Fatou, then it is complete. Theorem 11 appeared as the Nakano theorem in the following classical books: Peressini [Pe67], Thm 1.5, p. 142 and Fremlin [Fr74], Thm 23K, p. 50. Topological Riesz spaces satisfying both the Levi property and the Fatou property are called Nakano spaces (see Aliprantis-Burkinshaw [AB78], Def. 13.7, p. 94 and Aliprantis-Burkinshaw [AB03], Def. 4.35, p. 112). The conjunction of the Levi and Fatou properties is preserved by linear isometries in the class of Dedekind complete Banach lattices whose points are separated by order continuous functionals (see KantorovichAkilov [KA77], Thm 8, p. 388). 3d6. Weak compactness in order duals Nakano has an essential contribution to investigations of weak compactness in order duals. To study the deeper properties of the compact sets of En∼ we need to introduce the concept of order-equicontinuity. This fundamental notion appeared first in the work of Nakano under the term “universal-equicontinuity”: Let E be a Riesz space. A nonempty subset A of E ∼ is called order-equicontinuous on E, if 0 6 xn ↑6 x in E implies that (xn ) is a %A -Cauchy sequence where %A (y) = supf ∈A |f (x)|. The order-equicontinuity of A means exactly that the topology generated by a seminorm %A is a pre-Lebesgue topology. Monotone norms generating pre-Lebesgue topologies were called in Nakano’s terminology “strictly monotone norms” [Nb8, pp. 126–134] (see also AliprantisBurkinshaw [AB03], p. 75, historical note). Pre-Lebesgue topologies have a nice and very useful characterization which is due to Fremlin and Meyer-Nieberg: a locally solid topology τ is a pre-Lebesgue topology

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if and only if every order bounded disjoint sequence is τ -convergent to zero. It is a very deep and important result that in complete topological Riesz spaces the Lebesgue property and the pre-Lebesgue property coincide (see Aliprantis-Burkinshaw [AB03], Thm 3.24, p. 87). In particular, if we want to check order continuity of a norm in a concrete Banach lattice it is enough to verify that order bounded sequences with pairwise disjoint terms are norm null. Below we present Nakano’s criterion of ∗-weak compactness of subsets of En∼ ([Nb8, §28; see also Aliprantis-Burkinshaw [AB03], Thm 6.30, p. 155): THEOREM 12 (Nakano, 1950). Let E be an Archimedean Riesz space. Then for a nonempty subset A of En∼ the following statements are equivalent: (a) The set A is order-equicontinuous on E. (b) The solid hull of A is relatively σ(En∼ , E)-compact. (c) The seminorm %A is order continuous. Let us recall that the solid hull of a set A ⊂ E is defined as follows: sol A = {x ∈ E : |x| 6 |a| for some a ∈ A}. The set conv (sol A), called the convex-solid hull of A, is the smallest convex solid set containing A. Using his criterion Nakano proved also a Mazur-type theorem (see Aliprantis-Burkinshaw [AB03], Corollary 6.31, p. 156): THEOREM 13 (Nakano, 1950). Let E be a σ-Dedekind complete Riesz space. Then a subset of En∼ is relatively σ(En∼ , E)-compact if and only if its convex-solid hull is relatively σ(En∼ , E)-compact. The book of Nakano [Nb8] contains also the following important and useful result ([Nb8], Thm 19.6; see also Wnuk [Wn94]): If E is a σ-Dedekind complete Riesz space and (fn ) is a sequence of order bounded and order continuous functionals over E such that f (x) = limn→∞ fn (x) exists for all x ∈ E, then f is order continuous too.

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In other words, the above theorem says that En∼ is sequentially σ(E ∼ , E)-closed in E ∼ . H. H. Schaefer generalized Nakano’s result showing that every band in E ∼ is sequentially σ(E ∼ , E)-complete whenever E is σ-Dedekind complete. In the case of a Banach lattice E the order dual E ∼ coincides with the topological dual E ∗ (=the space of continuous linear functionals) and the sequential ∗-weak closedness of En∗ (=En∼ ) is the main argument in the proof that the (u)-property of PeÃlczy´ nski characterizes Banach lattices with order continuous norms 11 . Therefore the Nakano’s result has a beautiful application to a proof of a fundamental fact from the theory of Banach lattices: the order continuity of a norm is a topological invariant in a class of σ-Dedekind complete Banach lattices. Note that Nakano theorem is proved for the regular functionals not only for order continuous and Vulikh in his book [Vu67] presented this version of Nakano’s theorem together with its proof ([Vu67], Thm IX.1.1, pp. 245–247). 3e. Operators in Hilbert spaces Nakano proved several results, but let us present the von Neumann classical theorem on alternating orthogonal projections, which was also discovered by other authors, including Nakano. In 1933 John von Neumann [Ne33] (and also in 1949 [Ne49]) proved a theorem on convergence of the iterated product of orthogonal projections in Hilbert spaces, which was also rediscovered independently by Aronszajn [Ar50] in 1950, Nakano [Nb11] in 1953 and Wiener [Wi55] in 1955: Let A and B be two closed subspaces of a real Hilbert space H, and let PA : H → A and PB : H → B be the corresponding orthogonal projections of H onto A and B, respectively. Let x0 be an arbitrary point in H, and define the sequence of alternating projections by x2n+1 = PA x2n and x2n+2 = PB x2n+1 , where n = 0, 1, 2, . . . . Then the sequence {xn } converges in norm to P x0 , where P : H → M is the orthogonal projection of H onto the intersection M = A ∩ B. Nakano proved this theorem ([Nb11], Theorem 12.8, p. 48) in the 11 A Banach space X has the (u)-property of PeÃlczy´ nski if P for every weak Cauchy ∗ sequence (xn ) there exists another sequence (ynP ) such that ∞ n=1 |x (yn )| < ∞ for n ∗ ∗ all functionals x ∈ X and the sequence (xn − k=1 yk ) converges weakly to zero.

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following form: THEOREM 14 (Nakano, 1953). Let A and B be two closed subspaces of a Hilbert space H, and let PA : H → A and PB : H → B be the corresponding orthogonal projections of H onto A and B, respectively. Then limn→∞ (PA PB )n = PA∩B . Proof (Nakano [Nb12], pp. 48–49). Putting T = PB PA PB , we obtain a symmetric operator on H with its norm kT k ≤ 1. For an arbitrary a ∈ H, we hence have < T 2k a, T 2n a > = kT k+n ak2 , and, consequently, the sequence {< T 2k a, T 2n a >}∞ k=1 is convergent for every n = 1, 2, . . .. Then {T 2k a} is weakly convergent to some b ∈ H and kbk2 = = =

lim < T 2k a, b > = lim lim < T 2k a, T 2n a >

k→∞

lim lim < a, T

k→∞ n→∞

k→∞ n→∞ 2(k+n)

a > = < a, b >

lim < a, T 2k a >= lim kT 2k ak2 .

k→∞

k→∞

Therefore, the sequence {T 2k a} is strongly convergent, i.e., lim kT 2k a − bk = 0.

k→∞

Putting Q = limk→∞ T 2k we obtain a bounded linear operator Q on H. This operator is symmetric and idempotent since < Qx, y > = lim < T 2k x, y > = lim < x, T 2k y > =< x, Qy >, k→∞

k→∞

and Q2 = limk→∞ T 4k = Q. Thus Q is a projection operator. Since T 2 Qx = limk→∞ T 2k+2 x = Qx it follows that, for every x ∈ H, kQxk ≤ kT Qxk ≤ kPA PB Qxk ≤ kPB Qxk ≤ kQxk, and, hence, kQxk = kPB Qxk = kPA PB Qxk from which we conclude that Qx = PB Qx = PA PB Qx for every x ∈ H. Thus the range of Q is included in A ∩ B. On the other hand, for every x ∈ A ∩ B, we have that T x = PB PA PB x = x and so Q = PA∩B . Since T Q = PB PA PB Q = Q, we also have Q = limk→∞ T 2k+1 and,

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hence, Q = limk→∞ T k . On the other hand, we have obviously that (PA PB )k+1 = PA T k (k = 1, 2, . . .). Thus, Q = PA Q = lim PA T k = lim (PA PB )k . k→∞

k→∞

The proof is complete. Nakano also informed that this theorem was first published by himself in Japanese: On a theorem in the Hilbert space, Zenkoku Shijo Sugaku Danwakai 12 , Osaka 192(1940), 39–42 (see Nakano [Nb11], p. 298). The reviewer, Simeon Reich, of the paper Netyanun-Solmon [NS06] in MR2252935 (2007k:46039) wrote the following: While looking for a 1940 paper of Nakano [Zenkoku Shijo Sugaku Danwakai, Osaka 192 (1940), 39–42 ], which contains a proof of this result based on the spectral theorem for bounded self-adjoint operators, the authors of the present note came upon a related paper by S. Kakutani [Zenkoku Shijo Sugaku Danwakai, Osaka 192 (1940), 42–44 ], who proved this result by first noting that T = PB PA is asymptotically regular in the sense that {T k x − T k+1 x} → 0 as k → ∞ for each x ∈ H. Kakutani also stated that his proof can be extended to prove weak convergence when the iterates of PB PA are replaced by the iterated composition of any finite number of orthogonal projections. In the paper under review the authors show that Kakutani’s arguments can, in fact, be adapted to yield a proof of strong convergence in this case too. The von Neumann theorem was generalized to arbitrary finite number of projections by Halperin [Ha72] in 1972. An important special case is the iterative procedure of Kaczmarz [Ka37] (see also Maligranda [Ma07]) for solving large linear systems. On the other hand, if A and B are closed convex sets in H with nonempty intersection, then Br`egman [Br65] proved that the sequence {xn }, as in the von Neumann theorem, always converges weakly to some point in A ∩ B and for suitable sets A and B it even converges in the norm. In 2004 Hundal [Hu04] provides an 12

Zenkoku Shijo Sugaku Danwakai = Nationwide Seminar Reports on Mathematics.

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example of two closed convex subsets of a separable real Hilbert space l2 for which the method of alternating projection does not converge in norm. A good references to the history of the von Neumann theorem and the Br`egman theorem can be Netyanun-Solmon paper [NS06] and BauschkeMatouskov´a-Reich paper [BMR04], respectively. Of course, mathematicians then proceed to replace the orthogonal projections with any linear operators that are nonexpansive and nonnegative. Among the first authors were Amemiya and Ando [AA65], the students of Nakano. As a matter of fact, these results are susceptible to both linear and nonlinear generalizations, not only in Hilbert, but also in certain Banach spaces. F. Riesz-von Neumann-Miura proved that (see [RS90], Thm on p. 351): every closed linear transformation T with domain dense in separable Hilbert space H and such that T ^^A is a real or complex function of A. The symbol T ^^A means that every bounded symmetric transformation S : H → H which is permutable with A (SA = AS) is also permutable with T (ST = T S). Nakano also discuss this theorem in his papers [N39a], [N39c] and observe that this theorem in non-separable Hilbert space H is not necessarily true (cf. [RS90], p. 354). Let us mention that in the area of spectral theory of Hilbert space operators Nakano established a criterion of unitary equivalence of two normal operators and especially important was the case of nonseparable Hilbert space (Wecken and Nakano theory). P. R. Halmos in [Ha87] put the photo of Nakano (photo 257) and wrote the following: The first thing I learned about Nakano was his work on what is called the multiplicity theory of normal operators; that was something that had a great deal of interest for me once, and in my study of the subject I followed in the footsteps of Nakano (among others). Toward the end of his life he became interested in the foundations of set theory and distributed many preprints on that subject, but the professionals seemed to regard his approach with impatient suspicion. I would like to mention that Paul Richard Halmos (1916–2006) was John von Neumann’s assistant at the Institute for Advanced Study at

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Princeton when he became familiar with Nakano’s work. Osamu Takenouchi (Osaka University–student of Nakano) informed me that Nakano, after the war, was helping with the food for students and that Nakano was proud to say: von Neumann did a lot in the theory of Hilbert spaces but I did progress in the theory. Kazumi Nakano added, in the letter dated 18 October 2010, the following explanation: Takenouchi’s information, Nakano was helping with food for students, is difficult to understand without knowing the situation during April 1945 through March 1946 (the last year of the world war II). All educational institutions in Tokyo (kindergarten through university) were closed at the end of March 1945 and evacuated to safer regions. The Department of Mathematics of the Imperial University at Tokyo was evacuated to a local public school of village Osachi in Suwa region of Nagano prefecture. Shortage of food was everywhere. Nakano often led his students to hike for food-hunting in neighborhood villages. In those cold mountain areas of poor soil a sack potatoes would be a harvest of the day. His wife invited the students to the temple where the Nakano’s family maintained a temporary residence and fed hungry students. Nakano’s comment on von Neumann might be made during one of the hiking trips. Ando has mentioned the following anecdote ([An09], p. 2): when someone said to Nakano “You are von Neumann in Japan”, he replied immediately “No, von Neumann is Nakano in USA” It may not be well-known that Nakano corresponded to von Neumann frequently from 1939 to 1950 except the years of the war. John von Neumann (1903–1957), famous for his faithful response to letters he received, wrote back each time. Nakano whose hobby was photography set up equipment in his office at Hokkaido University to take pictures of von Neumann’s letters.

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Photo 13. Princeton, 8 August 1929. First letter of John von Neumann to Hidegoro Nakano

3f. Some other results We collect here some of Nakano’s results from a few subjects of his interest. • The Hahn-Banach theorem H. Nakano was interested in the extension of the classical HahnBanach theorem. He published papers on this question in 1957, 1959

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and 1971 ([N57], [N59b], [N71b]), where he replaced the subadditivity and positive homogeneity of the bounding function by its convexity. In the finite-valued case it even appeared in 1951 in his book [Nb11, § 79, Thm 2]. A convex functional on a real vector space X is a function p : X → R such that p(αx + (1 − α)y) ≤ α p(x) + (1 − α) p(y) for all x, y ∈ X and any 0 ≤ α ≤ 1. Of course, the Banach functional, that is, a subadditive and positively homegeneous functional p : X → R is a convex functional, and therefore the following theorem of Nakano includes the classical Hahn-Banach theorem. THEOREM 15 (Nakano, 1959). Let p be a convex functional on a real vector space X and let X0 be a vector subspace of X. If f0 be a linear functional on X0 such that f0 (x) ≤ p(x) for all x ∈ X0 , then there exists a linear functional f on X such that f (x) = f0 (x) for any x ∈ X0 and f (x) ≤ p(x) for any x ∈ X. A nice proof of this result can be found in the paper by Weston [We60], p. 445 and it is also in the Nakano book [Nb10, § 79, proof of Thm 2]. We should also mention here that Nakano [N59c] proved Theorem 15 even for the convex functional p : X → (−∞, +∞] satisfying the property that if p(tx) = ∞ for some x ∈ X and all t > 0, then p(tx) = ∞ for all t 6= 0. Musielak and Orlicz [MO60] found a simple proof of it by generalizing the Mazur-Orlicz theorem to convex functionals. Generalizations of the Nakano version of the Hahn-Banach extension theorem by modifying the conditions on the majorant functional p were given by Musielak and Orlicz [MO60], Nakano [N71b], Barac [Ba78] and Forg´ac [Fo76]. • Equivalent complete lattice norms Already Banach [Ba32] proved that two norms on a finite dimensional real vector space are equivalent. On the other hand, if (X, k · k1 ) is an infinite dimensional Banach space, then there exists a discontinuous linear functional f : X → R. Take u ∈ X such that f (u) = 1 and a discontinuous linear operator T : X → X defined by T x = x − 2 f (x) u.

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Then T 2 = I and kxk2 = kT xk1 defines a complete norm on X, which is not equivalent to the given norm k · k1 . Nakano proved in 1950 that all monotone Banach norms are equivalent (cf. [Nb10], Thm 30.28, p. 134). Recall that a norm k · k on a Riesz space E is said to be a monotone norm or a lattice norm if |x| ≤ |y| implies that kxk ≤ kyk. Since a positive operator from a Banach lattice to a normed Riesz space is continuous (see, for example, [AB85], Thm 12.3, p. 175) we obtain that all lattice norms that make a Riesz space a Banach lattice are equivalent. THEOREM 16 (Nakano, 1950). If two monotone norms k · k1 , k · k2 on a Riesz space E are both complete, then they are mutually equivalent, that is, there are constants a, b > 0 such that akxk1 ≤ kxk2 ≤ bkxk1 for all x ∈ E. Bukhvalov-Veksler-Lozanovski˘ı [BVL79, Thm 1.2, p. 164] refer to this theorem as Nakano’s theorem, but Aliprantis-Burkinshaw [AB85, Corollary 12.4, p. 176] and [AB, Corollary 5.22, p. 127] as Goffman’s theorem, which was proved by Goffman in 1956 (cf. Goffman [Go56], Thm 2). In the classical Kantorovich-Akilov book [KA77, Thm 2, p. 141] this theorem is without name. • Ergodic theorems In 1932 John von Neumann [Ne32] proved the mean ergodic theorem as a solution of a problem on unitary operators on Hilbert space: Let T be a unitary operator on a Hilbert space H (or, more generally, an isometric linear operator or only contraction), which is not necessarily surjective, that is, a linear operator satisfying kT xk = kxk for all x ∈ H, or, equivalently, satisfying T ∗ T = I, but not necessarily T T ∗ = I. Let P be the orthogonal projection onto {ϕ ∈ H : T (ϕ) = ϕ} = Ker(I − T ). Then, for any x ∈ H, we have n−1

1X k T (x) − P (x)kH = 0. lim k n→∞ n k=0

In the special case when H = L2 (Ω, Σ, µ) and T x(t) = x(τ (t)), where τ : Ω → Ω is a measure-preserving transformation, representing a time step of a discrete dynamical system. The ergodic theorem then means

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that the average behavior of a function x over sufficiently large time is approximated by the orthogonal component of x which is time-invariant. Generalizations to some Banach spaces and bounded linear operators were proved by F. Riesz (1938), K. Yosida (1938), S. Kakutani (1938), G. Birkhoff (1938) and E. R. Lorch (1939). For example, by approximation arguments it remains true for Lp (Ω, Σ, µ) with 1 < p < ∞, but it is not true for p = 1 or p = ∞ (cf. Krengel [Kr85], p. 4). In 1931 George David Birkhoff [Bi31] proved pointwise or strong ergodic theorem called also Birkhoff ergodic theorem (or sometimes Birkhoff-Khintchine ergodic theorem): let (Ω, Σ, µ) be a complete measure space and let τ : Ω → Ω be a measure-preserving transformation, that is, for any A ∈ Σ we have that τ −1 (A) ∈ Σ and µ(τ −1 (A)) = µ(A). If x ∈ L1 (Ω) (real or complex), then there exists a function x∗ ∈ L1 (Ω) such that n−1

1X x(τ k (t)) = x∗ (t), x∗ (τ (t)) = x∗ (t) for µ−almost all t ∈ Ω, n→∞ n lim

k=0

Pn−1 and kx∗ k1 ≤ kxk1 . If 0 < µ(Ω) < ∞, then limn→∞ k n1 k=0 x(τ k ) − ∗ x k1 = 0, and if in addition the measure-preserving transformation τ is ergodic (τ is ergodic if every invariant set A ∈ Σ, i.e., the set satisfying τ −1 (A) = A is such that either µ(A) = 0 or µ(Ω \ A) = 0), then x∗ is R 1 constant µ-a.e. on Ω and x∗ (t) = µ(Ω) Ω x(s) dµ µ-a.e. on Ω. After the classical ergodic theorems of von Neumann and Birkhoff, which both have sent in December 1931 their papers to the journal (Birkhoff even knew the von Neumann paper) it has been growing interest in extending these theorems in various directions. An operator P T : Lp → Lp has the pointwise ergodic property if, for 1 p k every x ∈ L , n n−1 k=0 T x(t) is a.e. convergent. This property holds if 1 < p < ∞, p 6= 2 and T is an isometry (Chacon-McGrath, 1969) or 1 < p < ∞ and T is a positive contraction (Akcoglou, 1975). Also the pointwise ergodic theorem is true for positive L1 − L∞ contraction T and x ∈ L1 (Dunford-Schwartz, 1956). The positivity of T , which is not needed in the von Neumann theorem is essential for a pointwise ergodic theorem in L2 . The counterxample in L2 was given by Burkholder (1962). I refer here to the paper by Feder [Fe81] and book by Krengel [Kr85].

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In 1948 H. Nakano [N48b], [49] proved an ergodic theorem in Banach lattices, as a generalization of the Birkhoff theorem. First, he introduced the notion of individual convergence for sequences in a Dedekind complete Riesz space and then he extended pointwise ergodic theorem to the lattice setting. In a Dedekind complete Riesz space E a sequence {xn } ⊂ E is called individually convergent to x ∈ E if lim [(xn ∧ y) ∨ z] = (x ∧ y) ∨ z for any y, z ∈ E.

n→∞

ind

Note that if Y is a universal completion of X, then xn → x in X if (o)

xn → x in Y (cf. Vulikh [Vu67], p. 144). In particular, the individual convergence in Lp , 1 ≤ p ≤ ∞, is a convergence almost everywhere since its universal completion is L0 and order convergence in L0 means convergence almost everywhere (see Vulikh [Vu67], p. 65). Moreover, (o)

ind

xn → x in X if and only if xn → x and the sequence {xn } is bounded in X. THEOREM 17 (Nakano, 1948). Let E be a Banach lattice with an order continuous and monotone complete norm. If T : E → E is a linear positive translator such that kT k k ≤ C P for all k = 1, 2, . . . for k some constant C > 0, then the sequence {[p] n1 n−1 k=0 T (x)} (n=1, 2, . . . ) converges individually for any invariant element p, that is, any p such that T (p) = p. We should collect notions used in the above theorem, which were not defined yet. A norm on E is called monotone complete (or Levi or universally monotone complete in the Nakano terminology) if for 0 ≤ xn %, supn∈N kxn k < ∞, then there exists x ∈ E such that xn % x. For a given W p ∈ E+ there corresponds to each x ∈ E+ a third element [p]x = [x ∧ (np) : n = 1, 2, . . .], which can be extended over E, and it defines a linear idempotent operator called projector. An operator T : E → E is a translator if T ([p] x) = [T p](T x) for all p, x ∈ E. Nakano’s ergodic theorem and individual convergence were extended or used, for example, by Cristescu [Cr58], Winkelbauer [Wi58] and Zaharopol [Za94].

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• Uniform spaces There are some properties defined on metric spaces but not on general topological spaces, for example, completeness or uniform continuity of functions. Generalizing metric spaces, A. Weil (1938) introduced the notion of uniform spaces. This notion can be defined in several ways and one such possibility was considered also by Nakano in 1968 in his book Uniform Spaces and Transformations Groups [Nb19]. About this book J. H. Williamson in the review [Wi71] wrote the following: This book had its origin in lecture course by the author at Wayne State University: the general level is about first-year postgraduate (English). (. . . ) What the author has done, in general terms, is to present in a different and more general setting a body of theory that is well known in the context of locally compact spaces and locally compact topological groups. He uses uniformities rather than topologies throughout, and replaces local compactness by “local total boundedness” of the uniformity, which has rather similar consequences. This enables him to formulate some familiar results in greater generality than customary: for example those on the existence and uniqueness of an invariant measure. He sets up the basic machinery for analysis on transformation groups and uses some of it in the last chapter to discuss almost periodic functions. The style of the book is somewhat austere; the development proceeds with few digressions in the way of illustrative examples or historical comments. • Group theory Nakano in a series of papers [N69], [N69a], [N72a] investigated the holomorph of a group. Let G be a group and let A(G) be its group of authomorphisms. The set H(G) of all pairs (g, σ), g ∈ G, σ ∈ A(G), forms a group under the composition (g, σ)(h, τ ) = (gσh, στ ).

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If e and I are the identity elements of G and A(G), respectively, then (e, I) is the identity of H(G). Furthermore, the inverse of (a, σ) is (σ −1 a−1 , σ −1 ). The group H(G) is called the holomorph of G. By identifying the element g ∈ G with the element (g, I) we obtain an imbedding of G in H(G). It is clear that G is an invariant subgroup of H(G) and that every authomorpism of G can be extended to an inner automorphism of H(G). Papers [N69] and [N69a] of Nakano were cited in the survey article written by Plotkin [Pl71], pp. 37, 41 or 544, 546 in the English translation. In 1972 Nakano [N72a] proved an equivalence theorem. Let G be the subgroup of H(G) of all right multiplication operations ra for a ∈ G and let Gl denote the subgroup of all left multiplications la : G → G for a ∈ G. THEOREM 18 (Nakano, 1972). Let M be a group. The following are equivalent: (i) there is a group G such that M is isomorphic to a subgroup M of H(G) containing G and Gl . (ii) there are subgroups R, L and B of M such that R E M , L E M , R = CM (L), L = CM (R), M = BR = BL and B ∩ R = B ∩ L = {e}. 13 In 1976 Hoffman [Ho76] gave a new proof of the Nakano theorem and added the third equivalent condition: there is a group G such that M is isomorphic to M ⊂ H(G) and G · I(G) ⊂ M . In this case, H(G) is considered as a split extension of G by A(G). Hoffman also shows that the conditions L E M and R = CM (R) of (ii) are redundant.

Acknowledgements. I would like to thank Prof. Kazumi Nakano, the daughter of Hidegoro Nakano, for the information about her father (cf. [Na6–9]), including several photos (photos 1–3, 5–9, 12, 13) and her corrections [Na10], to Prof. Tsuyoshi Ando for giving me his manuscript [An09] and photos 4, 11 of Nakano, for the titles of doctoral theses of 13

The symbol R E M means that R is a normal subgroup of M and CM (L) = {m ∈ M : b−1 mb = m for all b ∈ L} is a centralizer of L in M, which is a subgroup of M.

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Japanese students written under supervision of Nakano, for translation from Japanese into English of Nakano’s papers published in Functional Analysis, for the information on Yano’s books with translation of the preface from one of them and information about Japanese version of Yano’s article on Nakano [Ya84] together with its translation into English. I am also grateful to Prof. Witold Wnuk for the letter with his description of Nakano’s results in the theory of Riesz spaces on which part 3rd has been written, for the manuscript of his paper [Wn10] and for correcting my first version of the paper. Finally, I thank Prof. Mikio Kato (KIT) and Prof. Shinya Moritoh (NWU) for translating some documents connected with Nakano from Japanese into English, Dr. Yinru Hapu (Lule˚ a) for translation of the Kentaro Yano article [Ya84] about Nakano from Chinese into English and to the American Mathematical Society for permission to publish here photo 10, which is taken from the Halmos’ book [Ha87].

4

Books and text-books of Hidegoro Nakano

[Nb1]∗ Allgemeine Spektraltheorie: 1940–41, 1942, 382 pp. (in German); A collection of reprints published in various periodicals, 1940–1943. Includes bibliographical references and index 14 . [Nb2]∗ From Riemann integral to Lebesgue integral, Kangaekata-Kenkyusha, Tokyo 1940, 152 pp. (Japanese). [Nb3]∗ Hilbert Space Theory, Kyoritsu Publishers, Tokyo 1946, 248 pp. (Japanese). [Nb4−6]∗ Measure Theory, Shokabo, Tokyo, Vol. I 1947, 456 pp., Vol. II 1948, 868 pp., Vol. III 1950, 416 pp. (Japanese). [Nb7]∗ Classical Integration Theory, Kyoritsu Publishers, Tokyo 1949, 221 pp. (Japanese). [Nb8] Modulared Semi-Ordered Linear Spaces, Tokyo Math. Book Series, Vol. I, Maruzen, Tokyo 1950, i+288 pp. [Nb9] Modern Spectral Theory, Tokyo Math. Book Series, Vol. II, Maruzen, Tokyo 1950, vi+323 pp. [Nb10] Topology and Linear Topological Spaces, Tokyo Math. Book Series, Vol. III, Maruzen, Tokyo 1951, viii+281 pp. [Nb11] Spectral Theory in the Hilbert Space, Tokyo Math. Book Series, Vol. IV, Japan Society for the Promotion of Science, Tokyo 1953, iv+300 pp. 14

The sign star ∗, here and later on, mean that I didn’t see the paper, but I know about existence of it from other source.

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[Nb12]∗ Banach Space Theory, Tokai Shobo, Tokyo 1953, 213 pp. (Japanese). [Nb13] Semi-Ordered Linear Spaces, Japan Society for the Promotion of Science, Tokyo 1955, vi+508 pp. [Nb14]∗ Set Theory, Hokkaido Univ. Sapporo 1955, 60 pp. (Japanese). [Nb15]∗ Real Number Theory, Hokkaido Univ. Sapporo 1956, 76 pp. (Japanese). [Nb16]∗ Pedagogics of Mathematics (How to teach mathematics), Hokkaido Univ. Sapporo 1956, 90 pp. (Japanese). [Nb17]∗ Problems in Mathematics (for examination to universities), Hokkaido Univ. Sapporo 1956, 74 pp. (Japanese). [Nb18] Linear Lattices, Wayne State University Press, Detroit, Mich. 1966, 157 pp. (these are pages 9–153 from the book [Nb8]). [Nb19] Uniform Spaces and Transformation Groups, Wayne State University Press, Detroit, Mich. 1968, xv+253 pp. [Nb20]∗ Set Theory, Part I. Set theory, Part II. Collected papers on axiomatic set theory, Tokyo 1978, iv+243 pages, published posthumously by Hidegoro Nakano’s students.

5

List of scientific papers of Hidegoro Nakano

dy ¨ [N32] Uber den Konvergenzradius der L¨ osung einer Differentialgleichung dx = f (x, y), Proc. Imp. Acad. Jap. 8 (1932), 29–31. ¨ [N32a] Uber den Konvergenzradius der L¨ osungen eines Differentialgleichungssystems, Proc. Imp. Acad. Jap. 8 (1932), 113–115. ¨ [N32b] Uber die Verteilung der Peanoschen Punkte einer Differentialgleichung dy = f (x, y), Proc. Phys.-Math. Soc. Japan, III. Ser. 14 (1932), 41–43. dx ¨ [N32c] Uber eine stetige Matrixfunktion, Proc. Imp. Acad. Jap. 8 (1932), 217– 219. ¨ [N32d] Uber den Konvergenzbereich einer zweifachen Potenzreihe und seine Anwendungen, Jap. J. Math. 9 (1932), 135–144. ¨ [N32d] Uber die Verteilung der Nullstellen von den L¨ osungen der Differential2 gleichung ddzw2 + G(z)w = 0 einer komplexen Ver¨ anderlichen, Proc. Imp. Acad. Jap. 8 (1932), 337–339. [N34] Erweiterung des n-ten Mittelwertssatzes, Tohoku Math. J. 39 (1934), 200–201. [N34a] Zu den Abbildungen durch analytische Funktionen mehrerer komplexer Ver¨ anderlicher, Jap. J. Math. 11 (1934), 1–8. ¨ [N34b] Uber die Matrixfunktion, Jap. J. Math. 11 (1934), 9–13. ohnlichen linearen Differentialgleichungen, Jap. J. [N34c] Zur Theorie der gew¨ Math. 11 (1934), 59–129. [N34d] Zur Theorie der gew¨ ohnlichen Differentialgleichungen, J. Fac. Sci. Univ. Tokyo, Sect. I 3 (1934), 1–63.

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¨ [N35] Uber den Mittelwertsatz n-ter Ordnung, Proc. Phys.-Math. Soc. Japan, III. Ser. 17 (1935), 35–38. [N39] Zur Eigenwerttheorie normaler Operatoren, Proc. Phys.-Math. Soc. Japan, III. Ser. 21 (1939), 315–339. ¨ [N39a] Uber Abelsche Ringe von Projektionsoperatoren, Proc. Phys.-Math. Soc. Japan 21 (1939), 357–375. [N39b] Eine Bedingung f¨ ur die topologische Isomorphie von einem Banachschen Raum mit dem Hilbertschen, Jap. J. Math. 15 (1939), 287–296. [N39c] Funktionen mehrerer hypermaximaler normaler Operatoren, Proc. Phys.Math. Soc. Japan 21 (1939), 713–728. [N40] Hypermaximalit¨ at normaler Operatoren, Proc. Phys.-Math. Soc. Japan 22 (1940), 259–264. [N40a] Teilweise geordnete Algebra, Proc. Imp. Acad. Tokyo 16 (1940), 437–441. [N40b]∗ On a theorem in the Hilbert space, Zenkoku Shijo Sˆ ugaku Danwakai, Osaka 192 (1940), 39–42 (Japanese). [N41] Unit¨ arinvariante hypermaximale normale Operatoren, Ann. of Math. (2) 42 (1941), 657–664. ¨ [N41a] Uber den Beweis des Stoneschen Satzes, Ann. of Math. (2) 42 (1941), 665–667. [N41b] Teilweise geordnete Algebra, Jap. J. Math. 17 (1941), 425–511; reprinted in [Nb14], 1–87. [N41c] Eine Spektraltheorie, Proc. Phys.-Math. Soc. Japan (3) 23 (1941), 485– 511; reprinted in [Nb14], 88–114. [N41d] Unit¨ arinvarianten im allgemeinen Euklidischen Raum, Math. Ann. 118 (1941), 112–133. ¨ [N41e] Uber die Charakterisierung des allgemeinen C-Raumes, Proc. Imp. Acad. Tokyo 17 (1941), 301–307; reprinted in [Nb14], 125–131. ¨ [N41f] Uber das System aller stetigen Funktionen auf einem topologischen Raum, Proc. Imp. Acad. Tokyo 17 (1941), 308–310; reprinted in [Nb14], 115–117. ¨ [N41g] Uber normierte teilweisegeordnete Moduln, Proc. Imp. Acad. Tokyo 17 (1941), 311–317; reprinted in [Nb14], 118–124. ¨ [N41h] Uber Struktur von Spektren im allgemeinen Euklidischen Raum, Proc. Phys.-Math. Soc. Japan (3) 23 (1941), 871–882. ¨ [N42] Uber die Charakterisierung des allgemeinen C-Raumes. II, Proc. Imp. Acad. Tokyo 18 (1942), 280–286; reprinted in [Nb14], 132–138. [N42a] Stetige lineare Funktionale auf dem teilweisegeordneten Modul, J. Fac. Sci. Imp. Univ. Tokyo. Sect. I. 4 (1942), 201–382; reprinted in [Nb14], 139–321. [N42b] Riesz-Fischerscher Satz im normierten teilweise geordneten Modul, Proc. Imp. Acad. Tokyo 18 (1942), 350–353; reprinted in [Nb14], 323–326.

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¨ [N42c] Uber ein lineares Funktional auf dem teilweise geordneten Modul, Proc. Imp. Acad. Tokyo 18 (1942), 548–552; reprinted in [Nb14], 327–331. ¨ Erweiterungen von allgemein teilweisegeordneten Moduln. I., Proc. [N42d] Uber Imp. Acad. Tokyo 18 (1942), 626–630; reprinted in [Nb14], 332–336. ¨ [N43] Uber die Stetigkeit des normierten teilweise geordneten Moduls, Proc. Imp. Acad. Tokyo 19 (1943), 10–11; reprinted in [Nb14], 321–322. [N43a] Topologische Masse, Proc. Phys.-Math. Soc. Japan (3) 25 (1943), 279– 334. ¨ [N43b] Uber Erweiterungen von allgemein teilweisegeordneten Moduln. II., Proc. Imp. Acad. Tokyo 19 (1943), 138–143; reprinted in [Nb14], 337–342. ¨ [N44] Uber stochastischen Prozess. I, Proc. Imp. Acad. Tokyo 20 (1944), 513– 518. ¨ [N44a] Uber Einf¨ uhrung der teilweisen Ordnung im reellen Hilbertschen Raum, Proc. Phys.-Math. Soc. Japan (3) 26 (1944), 1–8; reprinted in [Nb14], 343–350. [N47] On the theory of Hilbert space. I. Bochner and Stone theorems, Sugaku (Mathematics) 1 (1947), no. 1, 38–39 (Japanese). [N47a] On the theory of Hilbert space. II. Spectral decomposition of bounded hermitian operators, Sugaku (Mathematics) 1 (1947), no. 1, 39–42 (Japanese). [N48] Reduction of Bochner’s theorem to Stone’s theorem, Ann. of Math. (2) 49 (1948), 279–280. [N48a] On the product of relative spectra, Ann. of Math. (2) 49 (1948), 281–315; reprinted in [Nb14], 351–385. [N48b] Ergodic theorems in semi-ordered linear spaces, Ann. of Math. (2) 49 (1948), 538–556; reprinted in [Nb14], 386–404. [N48c] Spectral theory and its application in continuous linear lattices in which a product is defined, Sugaku (Mathematics) 1 (1948), no. 2, 77–88 (Japanese). [N48d] On the theory of Hilbert space. III. Spectrum decomposition of normal operators, Sugaku (Mathematics) 1 (1947), no. 2, 97–101 (Japanese). [N49] The individual ergodic theorem in vector lattices, Sugaku (Mathematics) 1 (1949), no. 4, 257–263 (Japanese). [N50] Hilbert algebras, Tohoku Math. J. (2) 2 (1950), 4–23. [N51] Discrete semi-ordered linear spaces, Canad. J. Math. 3 (1951), 293–298 (with I. Halperin); reprinted in [Nb14], 405–410. [N51a] Modulared linear spaces, J. Fac. Sci. Univ. Tokyo. Sect. I. 6 (1951), 85– 131. [N51b] Modulared sequence spaces, Proc. Japan Acad. 27 (1951), 508–512; reprinted in [Nb14], 411–415. [N53] Linear topologies on semi-ordered linear spaces, J. Fac. Sci. Univ. Tokyo.

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Sect. I. 12 (1953), 87–104; reprinted in [Nb14], 416–433. [N53a] On transcendental points in proper spaces of discrete semi-ordered linear spaces, J. Fac. Sci. Univ. Tokyo. Sect. I. 12 (1953), 105–110; reprinted in [Nb14], 434–439. [N53b] Concave modulars, J. Math. Soc. Japan 5 (1953), 29–49; reprinted in [Nb14], 440–460. [N53c] Generalized lp spaces and the Schur property, J. Math. Soc. Japan 5 (1953), 50–58 (with I. Halperin). [N53d] Product spaces of semi-ordered linear spaces, J. Fac. Sci. Hokkaido Univ. Ser. I. 12 (1953), 163–210; reprinted in [Nb14], 461–508. [N53e] On completeness of uniform spaces, Proc. Japan Acad. 29 (1953), 490– 494. [N54] A generalization of Ascoli’s theorem, Proc. Japan Acad. 30 (1954), 282– 284. [N55] On the reflexivity of semi-continuous norms, Proc. Japan Acad. 31 (1955), 684–685 (with T. Mori and I. Amemiya). [N56] Modulars on semi-ordered linear spaces. I, J. Fac. Sci. Hokkaiko Univ. Ser. I. 13 (1956), 41–53 (with M. Miyakawa). [N57] An extension theorem, Proc. Japan Acad. 33 (1957), 603–604. [N59] Convergence concepts in semi-ordered linear spaces. I, Proc. Japan Acad. 35 (1959), 25–30 (with M. Sasaki). [N59a] Convergence concepts in semi-ordered linear spaces. II, Proc. Japan Acad. 35 (1959), 83–88. [N59b] On an extension theorem, Proc. Japan Acad. 35 (1959), 127. [N59c] On compactness of weak topologies, Proc. Japan Acad. 35 (1959), 444– 445. [N61] On unitary dilations of bounded operators, Acta Sci. Math. Szeged 22 (1961), 286–288. [N63] Monotone functions on linear lattices, Canad. J. Math. 15 (1963), 226– 236 (with H. W. Ellis). [N65] Invariant metrics, Math. Ann. 162 (1965), 89–91. [N66] Outer measures on a linear lattice, Trans. Amer. Math. Soc. 122 (1966), 277–288 (with L. Brown). [N66a] A representation theorem for Archimedean linear lattices, Proc. Amer. Math. Soc. 17 (1966), 835–837 (with L. Brown). [N66b] Quasi-norm spaces, Trans. Amer. Math. Soc. 123 (1966), 1–31 (with R. Metzler). [N67] Semi-invariant measures, Math. Ann. 172 (1967), 247–248. [N67a] Quasi-bounded linear lattices, J. London Math. Soc. 42 (1967), 577–590 (with B. J. Eisenstadt). [N68] Generalized modular spaces, Studia Math. 31 (1968), 439–449.

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[N68a] Critical modulars, Studia Math. 31 (1968), 451–453. [N69] Transformation groups on a group, Math. Ann. 181 (1969), 81–96. [N69a] Representations of a group by tranformations on its subgroups, Math. Ann. 181 (1969), 173–180. [N69b] On the Haar measures of subgroups, Math. Ann. 181 (1969), 285–287. [N70] An axiomatic set theory, Math. Ann. 186 (1970), 53–64. [N70a] Pointless axiomatic set theory, Math. Ann. 186 (1970), 271–281. [N71] Semi-continuous linear lattices, Studia Math. 37 (1971), 191–195 (with B. C. Anderson). [N71a] Cluster lattices, Bull. Acad. Polon. Sci. S´er. Sci. Math. Astronom. Phys. 19 (1971), 5–7 (with S. Romberger). [N71b] On the Hahn-Banach theorem, Bull. Acad. Polon. Sci. S´er. Sci. Math. Astronom. Phys. 19 (1971), 743–745. [N72] On semicontinuous linear lattices, Proc. Amer. Math. Soc. 34 (1972), 115–117. [N72a] Automoration groups, Proc. London Math. Soc. (3) 24 (1972), 171–191. [N73] Generalized relative spectra, Comment. Math. Prace Mat. 17 (1973), 29–34 (with J. E. Brierly). [N73a] Discrete linear lattices, Comment. Math. Prace Mat. 17 (1973), 179–185 (with R. G. Mosier). [N74] Simple linear lattices, Comment. Math. Prace Mat. 18 (1974), 141–149 (with H. H. Chuang). [N74a] On distributive lattices, Comment. Math. Prace Mat. 18 (1974), 151–153 (with S. Romberger). [N75] Lp, q modulars, Proc. Amer. Math. Soc. 50 (1975), 201–204. [N75a] Connector theory, Pacific J. Math. 56 (1975), no. 1, 195–213 (with K. Nakano). [N81] Double integral theorem of Haar measures, Hokkaido Math. J. 10 (1981), no. 2, 183–208.

Abstracts of Nakano [Na65a] (with L. Brown) A representation theorem for Archimedean linear lattice (15 Feb. 1965), Notices Amer. Math. Soc. 12 (1965), 332. [Na65b] Invariant metrics (27 April 1965), Notices Amer. Math. Soc. 12 (1965), 451. [Na65c] Transformation groups on a group (27 Sept. 1965), Notices Amer. Math. Soc. 12 (1965), 794. [Na66] Mathematical set theory (20 Jan. 1966), Notices Amer. Math. Soc. 13 (1966), 326. [Na67] On the continuum problem (21 Aug. 1967), Notices Amer. Math. Soc. 14 (1967), 815.

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[Na68a] On the existence of regular initial numbers (15 Jan. 1968), Notices Amer. Math. Soc. 15 (1968), 472. [Na68b] Representation of a group by transformations on its subgroups (17 Jan. 1968), Notices Amer. Math. Soc. 15 (1968), 506. [Na68c] Critical numbers (9 April 1968), Notices Amer. Math. Soc. 15 (1968), 724. [Na68d] Mapping law (12 June 1968), Notices Amer. Math. Soc. 15 (1968), 1030. [Na69a] Elimination of the paradoxes in the set theory (24 June 1968), Notices Amer. Math. Soc. 16 (1969), 88. [Na69b] An axiomatic set theory (18 Oct. 1968), Notices Amer. Math. Soc. 16 (1969), 500. [Na69c] Pointless axiomatic set theory (23 Oct. 1968), Notices Amer. Math. Soc. 16 (1969), 533. [Na69d] The Russell’s paradox in set spaces (Axiomatic set theory is not set theory) (5 May 1969), 16 (1969), 764. [Na69e] (with R. Mosier) Discrete linear lattices (24 Sept. 1969), 16 (1969), 1063. [Na69f] (with R. Romberger) Cluster lattices (29 Sept. 1969), 16 (1969), 1063. [Na69g] (with B. Andersson) Semi-continuous linear lattice (30 Sept. 1969), 16 (1969), 1063. [Na70a] Setology (14 Oct. 1969), 17 (1970), 138. [Na70b] What is axiomatic set theory ? (24 Nov. 1969), 17 (1970), 393. [Na71a] Automoration groups (4 Sept. 1970), 18 (1971), 127. [Na71b] Lp modulars (16 Feb. 1971), 18 (1971), 517. [Na71c] On characterization of automorphism groups (6 May 1971), 18 (1971), 765. [Na71d] On the Hahn-Banach theorem (11 Jan. 1971), 18 (1971), 926. [Na72a] Double integral theorem of Haar measures (30 Aug. 1971), 19 (1972), A-96. [Na72b] Automoration groups, Zentralblatt f¨ ur Math. 236 (1972), 20031. [Na73] Representation theory of transformation groups (7 Sept. 1972), 20 (1973), A-87. [Na74] Correct set theory (30 Oct. 1973), 21 (1974), A-29.

Papers of Nakano in Japanese journal “Functional Analysis” In 1947 Nakano Laboratory at the Department of Mathematics, Faculty of Science, University of Tokyo started the journal Functional Analysis in Japanese language (at present we will rather say technical reports). Volume I of 219 pages was published in years 1947–1949, Vol. II of 175 pages in 1949–1952 and Vol. III of 39 pages in 1954. I didn’t see these papers but thanks to Prof. Ando I know the contents and will give

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the list of Nakano’s papers (all published in Japanese language). Except Nakano there were also another authors of papers: Amemiya, Ito, Iwata, Koshi, Kuroda, Miyatake, Miyazaki, Mori, Shibata, Takenouchi and Yamamuro. Some papers were then published in different journals in English language. Most of content of Nakano’s papers [N1], [N2], [N3]– [N6], [N8], [N9], [N12], [N13] and [N14] are incorporated in his book [Nb8]. [N1]∗ Modulared semi-ordered linear spaces, Functional Analysis I (1947–49), no. 1, 1–40 (Rec. 1947/9/10). [N2]∗ Reflexive extension of semi-ordered linear spaces, Functional Analysis I (1947–49), no. 2, 41–78 (Rec. 1948/2/20). [N3]∗ Calculation of projections, Functional Analysis I (1947–49), no. 2, 79–86 (Rec. 1948/2/20). [N4]∗ Conjugately similar semi-ordered linear spaces, Functional Analysis I (1947–49), no. 3, 87–106 (Rec. 1948/4/14). [N5]∗ Superuniversally continuous semi-ordered linear spaces, Functional Analysis I (1947–49), no. 3, 106–110 (Rec. 1947/10/5). [N6]∗ Totally continuous semi-ordered linear spaces, Functional Analysis I (1947-49), no. 3, 111–126 (Rec. 1947/10/5). [N7]∗ Modulared spaces, Functional Analysis I (1947–49), no. 4, 127–147 (Rec. 1948/5/20); English version published in [N51a]. [N8]∗ On quotient systems of normed linear spaces, Functional Analysis I (1947-49), no. 4, 148–150 (Rec. 1948/5/20). [N9]∗ On the conjugate norm of a uniformly convex norm, Functional Analysis I (1947–49), no. 4, 151–152 (Rec. 1948/6/20). [N10]∗ A simple proof of ℵω α = ℵα , Functional Analysis I (1947–49), no. 4, 153–158 (Rec. 1948/5/25). [N11]∗ Calculation of projections. II, Functional Analysis I (1947–49), no. 4, 156–158 (Rec. 1948/6/20). [N12]∗ Modulared semi-ordered linear spaces II, Functional Analysis I (1947– 49), no. 5, 159–170 (Rec. 1948/6/20). [N13]∗ Divergent sequences in semi-ordered linear spaces, Functional Analysis I (1947–49), no. 5, 171–175 (Rec. 1948/5/20). [N14]∗ Separability of semi-ordered linear spaces, Functional Analysis I (1947– 49), no. 6, 187–203 (Rec. 1948/7/20). [N15]∗ Discrete semi-ordered linear spaces, Functional Analysis I (1947–49), no. 6, 204–207 (Rec. 1948/9/20); English version published in [N51]. [N16]∗ Hilbert algebras, Functional Analysis II (1947–49), no. 1, 1–19 (Rec. 1949/3/20); English version published in [N50]. [N17]∗ Product of semi-ordered linear spaces, Functional Analysis II (1949–52),

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no. 2, 37–61 (Rec. 1949/6/20); English version published in [N53d]. [N18]∗ Modulared sequence spaces, Functional Analysis II (1949–52), no. 3, 69– 73 (Rec. 1951/2/25); English version published in [N51b]. [N19]∗ Concave modulared semi-ordered linear spaces, Functional Analysis II (1949–52), no. 4, 91–98 (Rec. 1951/2/25); English version published in [N53b]. [N20]∗ On continuity of quasi-norms, Functional Analysis II (1949–52), no. 4, 99–100 (Rec. 1951/9/25). [N21]∗ Topological semi-ordered linear spaces, Functional Analysis II (1949–52), no. 5, 125–137 (Rec. 1952/2/25); English version published in [N53]. [N22]∗ On transcendental points in discrete semi-ordered linear spaces, Functional Analysis II (1949–52), no. 6, 153–157 (Rec. 1952/9/30); English version published in [N53a]. [N23]∗ On quasi-continuous functions, Functional Analysis III (1954), no. 1, 28–30 (Rec. 1954/9/29).

6

Information about Hidegoro Nakano

[An09] T. Ando, Personal reminiscences on Professor Hidegoro Nakano and his school, manuscript, June 26, 2009, 3 pages. [Ma06a] L. Maligranda, Hidegoro Nakano (1909–1974)–on the centenary of his birth, Abstract of the lecture at The Third International Symposium “Banach and Function Spaces 2009”, Sept. 14–17, 2009, Kyushu Institute of Technology, Kitakyushu, Japan, page 5. [Ma06b] L. Maligranda, Hidegoro Nakano (1909–1974)–on the centenary of his birth, Lecture delivered on the 14 September 2009 at The Third International Symposium “Banach and Function Spaces 2009”, Sept. 14–17, 2009, Kyushu Institute of Technology, Kitakyushu, Japan, 90 pages. [Na93] T. Nakazi, Biography of T. Ando, in: Contributions to Operator Theory and its Applications, The Tsuyoshi Ando Anniversary Volume, Edited by T. Furuta, I. Gohberg, T. Nakazi, Oper. Theory Adv. Appl. 62, Birkh¨auser, Basel 1993, 1–10. [Na69] H. Nakano, Letter to WÃladysÃlaw Orlicz with 7 abstracts, Detroit, 7 Jan. 1969, 7 pages, Archive of the Polish Academy of Sciences in Pozna´ n, sign. P. III 91. [Na6–9] K. Nakano, Letters to Lech Maligranda with information about Hidegoro Nakano, Brockport, NY, 24 June 2006 (1 page), 23 Aug. 2006 (2 pages), 20 Sept. 2006 (2 pages), 8 June 2008 (4 pages), 19 Nov. 2008 (2 pages), 21 April 2009 (4 pages). [Na10] K. Nakano, Letter to Lech Maligranda with corrections in his paper on Hidegoro Nakano, Brockport, NY, 10 October 2010 (1+5 pages+copy of

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the letter of J. von Neumann to H. Nakano from 1939). [Ya84] K. Yano, Strange Mathematicians, No. 13, Hidegoro Nakano (1909– 1974), Shincho Publishing Company, Tokyo 1984, 125–134 (Japanese) 15 ; Chinese translation by Yan Yi-Qing, Excellent Mathematicians, No. 13, Nakano Hidegoro (1909–1974), in: Mathmedia (e-version; Chinese edition) 17, 1 (1993), 1–3 at http://www.math.sinica.edu.tw/media/

7

Books and papers connected with Nakano’s results

[Ab02] Ju. A. Abramovich, Weakly compact sets in topological K-spaces, Teor. Funkci˘ı Funkcional. Anal. i Prilozhen. 15 (1972), 27–35 (Russian). [AA02] Y. A. Abramovich and C. D. Aliprantis, An Invitation to Operator Theory, Graduate Studies in Math. 50, AMS, Providence, RI 2002 [Nakano, p. 27, 95, 205, 516]. [AA02a] Y. A. Abramovich and C. D. Aliprantis, Problems in Operator Theory, Graduate Studies in Math. 51, AMS, Providence, RI 2002 [Nakano, p. 31, 156, 377]. [AB75] C. D. Aliprantis and O. Burkinshaw, A new proof of Nakano’s theorem in locally solid Riesz spaces, Math. Z. 144 (1975), no. 1, 25–33. [AB76] C. D. Aliprantis and O. Burkinshaw, Nakano’s theorem revisited, Michigan Math. J. 23 (1976), no. 2, 173–176. [AB78] C. D. Aliprantis and O. Burkinshaw, Locally Solid Riesz Spaces, Academic Press 1978 [Nakano, p. 10, 23, 26, 37, 53, 63, 73, 81, 93–95, 147, 193]. [AB85] C. D. Aliprantis and O. Burkinshaw, Positive Operators, Academic Press, New York 1985 [Nakano, p. 56, 59, 60, 73, 191–192, 357]. 15

Kentaro Yano (1912–1993) was a famous Japanese geometer (differential geometry). He published many illuminating mathematical books for general audience. Among them, there are three pocket books of episodes of celebrated mathematicians, published by Shincho Publishing Company (SPC) and whose literal translations are: Excellent Mathematicians, SPC, Tokyo 1980, Delightful Mathematicians, SPC, Tokyo 1981 and Strange Mathematicians, SPC, Tokyo 1984. The information on Nakano is in the third one. The third book contains episodes of 24 mathematicians. It is not a collection of 24 articles published earlier somewhere, but Yano wrote all specially for the pocket book. The book was published in 1984/6/25 and the name of pocket book series is “Shincho Bunko”. In the preface of the third book Yano writes: Yes, episodes of mathematicians are usually a little curious. As mathematicians think on mathematics intently, outside of mathematics their behaviors are sometimes strange. It is without doubt that the more strange their behaviors are the better their mathematics is.

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[AB03] C. D. Aliprantis and O. Burkinshaw, Locally Solid Riesz Spaces with Applications to Economics, Mathematical Surveys and Mongraphs vol. 105, Amer. Math. Soc. 2003 [Nakano, p. 19, 24, 25]. [AB06] C. D. Aliprantis and O. Burkinshaw, Positive Operators, Second Ed., Springer 2006 [Nakano, p. 63, 111, 130, 186]. [Am61] I. Amemiya, On ordered topological linear spaces, in: 1961 Proc. Internat. Sympos. Linear Spaces (Jerusalem, 1960), Jerusalem Academic Press, Jerusalem; Pergamon, Oxford 1961, 14–23. [AA65] I. Amemiya and T. Ando, Convergence of random products of contractions in Hilbert space, Acta Sci. Math. (Szeged) 26 (1965), 239–244. [An59] T. Ando, On the continuity of the norm by a modular, Fund. Appl. Aspects Math. 1 = Res. Inst. Appl. Electricity Monograph, Hokkaido Univ. 7 (1959), 31–44. [An59a] T. Ando, Convexity and evenness in modulared semi-ordered linear spaces, J. Fac. Sci. Hokkaido Univ. Ser. I 14 (1959), 59–95. [Ar50] N. Aronszajn, Theory of reproducing kernels, Trans. Amer. Math. Soc. 68 (1950), 337–404 [Ba32] S. Banach, Th´eorie des Op´erations Lin´eaires, Monografje Matematyczne I, Warszawa 1932, viii+252 pp.; Reprinted in: Stefan Banach, Oeuvres, Vol. II, PWN, Warszawa 1979, 13–219. [Ba78]∗ D. Barac, Sur le prolongement des fonctionnelles major´ees par une fonctionnelle sous-additive, Studia Univ. Babes-Bolyai Math. 23 (1978), no. 1, 55–60. [BMR04] H. H. Bauschke, E. Matouskov´a and S. Reich, Projections and proximal point methods: Convergence results and counterexamples, Nonlinear Anal. 56 (2004), 715–738. [Be79] S. J. Bernau, Review of the book Locally solid Riesz spaces, by Charalambos D. Aliprantis and Owen Burkinshaw, Bull. Amer. Math. Soc. (N.S.) 1 (1979), no. 6, 997–1003. [Bi31] G. D. Birkhoff, Proof of the ergodic theorem, Proc. Nat. Acad. Sci. USA 17 (1931), 656–660. [BG01] O. Blasco and P. Gregori, Type and cotype in vector-valued Nakano sequence spaces, J. Math. Anal. Appl. 264 (2001), no. 2, 657–672. [Bo40] H. F. Bohnenblust, An axiomatic characterization of Lp -spaces, Duke Math. J. 6 (1940), 627–640. [Br65] L. M. Br`egman, The method of successive projection for finding a common point of convex sets, Dokl. Akad. Nauk SSSR 162 (1965), 487–490; English transl. in: Sov. Math. Dokl. 6 (1965), 688–692. [BVL79] A. V. Bukhvalov, A. I. Veksler and G. Ja. Lozanovski˘ı, Banach lattices — some Banach aspects of the theory, Uspekhi Mat. Nauk 34 (1979), no. 2(206), 137–183; English transl. in: Russian Math. Surveys 34 (1979),

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no. 2, 159–212. [BD76] O. Burkinshaw and P. Dodds, Weak sequential compactness and completeness in Riesz spaces, Canad. J. Math. 28 (1976), no. 6, 1332–1339. [CK98] P. G. Casazza and N. J. Kalton, Uniqueness of unconditional bases in Banach spaces, Israel J. Math. 103 (1998), 141–175. ` propos des th´eor`emes ergodiques individuels de H. Naka[Cr58] R. Cristescu, A no, An. Univ. “C. I. Parhon” Bucuresti. Ser. Sti. Nat. 7 (1958), no. 20, 23–25. [Dh00] S. Dhompongsa, Convexity properties of Nakano spaces, Science Asia 26 (2000), 21–31. [DHN05] L. Diening, P. H¨ast¨o and A. Nekvinda, Open problems in variable exponent Lebesgue and Sobolev spaces, in: Function Spaces, Differential Operators and Nonlinear Analysis (FSDONA04), Proc. Int. Conference Milovy, Czech Rep., May 28-June 4, 2004, Math. Inst. Acad. Sci. Czech Rep., Prague 2005, 38–58. [DU77] J. Diestel and J. J. Uhl, Jr., Vector Measures, AMS, Providence 1977, xiii+322 pp. [Do75] P. G. Dodds, Sequential convergence in the order duals of certain classes of Riesz spaces, Trans. Amer. Math. Soc. 203 (1975), 391–403. [DO68] L. Drewnowski and W. Orlicz, A note on modular spaces. X, Bull. Acad. Polon. Sci. S´er. Sci. Math. Astronom. Phys. 16 (1968), 809–814. [DS58] N. Dunford and J. T. Schwartz, Linear Operators. I. General Theory, Interscience Publishers, New York, London 1958, xiv+858 pp. [Fa07] X. L. Fan, Amemiya norm equals Orlicz norm in Musielak-Orlicz spaces, Acta Math. Sin. (Engl. Ser.) 23 (2007), no. 2, 281–288. [Fe81] M. Feder, On power-bounded operators and the pointwise ergodic property, Proc. Amer. Math. Soc. 83 (1981), no. 2, 349–353. [FJ03] R. Fleming and J. E. Jamison, Isometries on Banach Spaces: Function Spaces, Chapman & Hall, Boca Raton 2003. x+197 pp. [Nakano, p. 141, 144, 189]. [Fo76] L. Forg´ac, On the Hahn-Banach theorem, Math. Slovaca 26 (1976), no. 1, 39–45. [Fr74] D. H. Fremlin, Topological Riesz Spaces and Measure Theory, Cambridge University Press, London-New York 1974, xiv+266 pp. [Nakano, p. 44, 50, 53, 138, 172, 259]. [Go56] C. Goffman, Compatible seminorms in a vector lattice, Proc. Nat. Acad. Sci. U.S.A. 42 (1956), 536–538. [Go59] C. Goffman, Completeness in topological vector lattices, Amer. Math. Monthly 66 (1959), 87–92. [Ha87] P. R. Halmos, I have a photographic memory, AMS, Providence 1987, x+326 pp. [Nakano, picture 257].

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[Ha72] I. Halperin, The product of projection operators, Acta Sci. Math. (Szeged) 23 (1962), 96–99. [HN53] I. Halperin and H. Nakano, Generalized lp spaces and the Schur property, J. Math. Soc. Japan 5 (1953), 50–58. [HM00] H. Hudzik and L. Maligranda, Amemiya norm equals Orlicz norm in general, Indag. Math. (N.S.) 11 (2000), no. 4, 573–585. [HWY94] H. Hudzik, C. Wu and Y. N. Ye, Packing constant in Musielak-Orlicz sequence spaces equipped with the Luxemburg norm, Rev. Mat. Univ. Complut. Madrid 7 (1994), no. 1, 13–26. [Hu04] H. Hundal, An alternating projection that does not converge in norm, Nonlinear Anal. 57 (2004), no. 1, 35–61. [Is59] J. Ishii, On equivalence of modular function spaces, Proc. Japan Acad. 35 (1959), 551–556. [Is59] J. Ishii and T. Shimogaki, On Haar functions in the space LM (ξ, t) , J. Fac. Sci. Hokkaido Univ. Ser. I 17 (1963), 55–63. [It61] T. Ito, A generalization of Mazur-Orlicz theorem on function spaces, J. Fac. Sci. Hokkaido Univ. Ser. I 15 (1961), 221–232. [Ju41] A. I. Judin, On the extension of partially ordered linear spaces, Uch. Zap. Leningrad State Univ. 13 (1941), 57–61 (Russian). [Ka37] S. Kaczmarz, Angen¨ aherte Auflsung von Systemen linearer Gleichungen, Bull. Int. Acad. Pol. Sci. Lett. A. 1937, 355–357. [Ka40]∗ S. Kakutani, On Nakano’s talk, Zenkoku Shijo Sugaku Danwakai, Osaka 192 (1940), 42–44 (Japanese). [Ka41] S. Kakutani, Concrete representation of abstract (L)-spaces and the mean ergodic theorem, Ann. of Math. 42 (1941), 523–537. [Ka61] J. A. Kalman, Some problems related to H¨ older’s inequality, Proc. London Math. Soc. (3) 11 (1961), 311–326. [Ka61] A. Kami´ nska, Uniform rotundity of Musielak-Orlicz sequence spaces, J. Approx. Theory 47 (1986), no. 4, 302–322. [KM02] A. Kami´ nska and M. MastyÃlo, The Schur and (weak) Dunford-Pettis properties in Banach lattices, J. Aust. Math. Soc. 73 (2002), no. 2, 251–278. [KA77] L. V. Kantorovich and G. P. Akilov, Functional Analysis, Nauka, Moskwa 1977 [Nakano, p. 255, 380, 385]. [Ka70] S. Kaplan, On weak compactness in the space of Radon measures, J. Funct. Anal. 5 (1970), 259–298. [Ka07] A. Yu. Karlovich, Algebras of singular integral operators with piecewise continuous coefficients on weighted Nakano spaces, Oper. Theory Adv. Appl. 171, Birkh¨auser, Basel 2007, 171–188. [Ka98] E. Katirtzoglou, Type and cotype of Musielak-Orlicz sequence spaces, J. Math. Anal. Appl. 226 (1998), no. 2, 431–455.

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[Kl65] V. Klee, Summability in l(p1 , p2 , · · · ) spaces, Studia Math. 25 (1965), 277–280. [KR58] M. A. Krasnosel’ski˘ı and Ja. B. Ruticki˘ı, Vypuklye Funktsii i Prostranstva Orlicza, Gosudarstv. Izdat. Fiz.-Mat. Lit., Moscow 1958, 271 pp.; English transl. Convex Functions and Orlicz Spaces, Noordhoff Ltd., Groningen 1961, xi+249 pp. [Kr85] U. Krengel, Ergodic Theorems, Walter de Gruyter& Co., Berlin 1985, viii+357 pp. [La74] E. H. Lacey, The Isometric Theory of Classical Banach Spaces, SpringerVerlag 1974. [LS68] V. I. Levchenko and I. V. Shragin, Nemychi˘ı’s operator acting from the space of continuous functions into an Orlicz-Nakano space, Mat. Issled. 3 (1968), vyp. 3 (9), 91–100 (Russian). [LT77] J. Lindenstrauss and L. Tzafriri, Classical Banach spaces. I. Sequence spaces, Springer-Verlag, Berlin-New York 1977. [LT79] J. Lindenstrauss and L. Tzafriri, Classical Banach Spaces. II. Function Spaces, Springer-Verlag, Berlin-New York 1979 [Nakano, p. 4, 27, 236]. [Lo71] G. Ja. Lozanovski˘ı, Normed lattices with a semicontinuous norm, Sibirsk. Mat. Z. 12 (1971), 232–234; English transl. in Siberian Math. J. 12 (1971), 169–170. [Lo73] G. Ja. Lozanovski˘ı, The second Nakano-dual space to a Banach lattice, Optimizacija Vyp. 12(29) (1973), 90–92 (Russian). [Lu55] W. A. J. Luxemburg, Banach function spaces, Thesis, Technische Hogeschool te Delft 1955, 71 pp. [Nakano, p. 2, 19, 49–50, 54, 65, 69]. [LZ56] W. A. J. Luxemburg and A. C. Zaanen, Conjugate spaces of Orlicz spaces, Indag. Math. 18 (1956), 217–228. [LZ63] W. A. J. Luxemburg and A. C. Zaanen, Notes on Banach function spaces. I, Indag. Math. 25 (1963), no. 2, 135–147. [LZ71] W. A. J. Luxemburg and A. C. Zaanen, Riesz Spaces I, North-Holland, Amsterdam 1971 [Nakano, p. 191, 287–288, 341, 347–348, 355–356, 429, 509–510]. [Ma89] L. Maligranda, Orlicz Spaces and Interpolation, Seminars in Mathematics 5, University of Campinas, Campinas SP 1989, iii+206 pp. [Nakano, p. 1, 6, 202]. [Ma98] L. Maligranda, Why H¨ older’s inequality should be called Rogers’ inequality, Math. Inequal. Appl. 1 (1998), 69–83. [Ma07] L. Maligranda, Stefan Kaczmarz (1895–1939), Antiq. Math. 1 (2007), 15 – 61 (Polish); Also at: http: //leksykon.ptm.mimuw.edu.pl / biogramy/kaczmarz/kaczmarz.php [LM08] L. Maligranda, Tosio Aoki (1910–1989), in: Banach and Function Spaces II, Proc. of the Internat. Symp. on Banach and Function Spaces (IS-

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[NS06] [Ne32] [Ne33]

[Ne49] [NP81] [Or31]

[Or61]

L. Maligranda BFS2006) (Sept. 14–17, 2006, Kitakyushu-Japan), (Editors M. Kato and L. Maligranda), Yokohama Publishers 2008, 1–23. S. Mazur and W. Orlicz, Sur les espaces m´etriques lin´eaires. I, Studia Math. 10 (1948), 184–208; Reprinted in: WÃladysÃlaw Orlicz, Collected Papers, PWN, Warszawa 1988, 557–581. S. Mazur and W. Orlicz, Sur les espaces m´etriques lin´eaires. II, Studia Math. 13 (1953), 137–179; Reprinted in: WÃladysÃlaw Orlicz, Collected Papers, PWN, Warszawa 1988, 671–713. A. A. Mekler and N. F. Sokolovskaya, Subspaces in Nakano’s sense of conditionally complete vector lattices, Functional Analysis, Ulyanovsk. Gos. Ped. Inst., Ulyanovsk, No. 18 (1982), 92–101 (Russian). P. Meyer-Nieberg, Banach Lattices, Springer-Verlag, Berlin 1991, xvi+395 pp. [Nakano, p. 32, 37, 38, 40, 388]. J. Musielak, Modular Spaces, Wyd. Naukowe UAM, Pozna´ n 1978, 136 pp. (Polish). J. Musielak, Orlicz Spaces and Modular Spaces, Lecture Notes in Mathematics 1034, Springer-Verlag, Berlin 1983, iii+222 pp. [Nakano, p. 7, 164–166, 168–170, 173, 183, 186, 189, 210]. J. Musielak and W. Orlicz, On modular spaces, Studia Math. 18 (1959), 49–65; Reprinted in: WÃladysÃlaw Orlicz, Collected Papers, PWN, Warszawa 1988, 1052–1068. J. Musielak and W. Orlicz, A generalization of certain extension theorems, Bull. Acad. Polon. Sci. S´er. Sci. Math. Astronom. Phys. 8 (1960), 531–534; Reprinted in: WÃladysÃlaw Orlicz, Collected Papers, PWN, Warszawa 1988, 1117–1120. A. Netyanun, and D. Solmon, Iterated products of projections in Hilbert space, Amer. Math. Monthly 113 (2006), no. 7, 644–648. J. von Neumann, Proof of the quasi-ergodic hypothesis, Proc. Nat. Acad. Sci. USA 18 (1932), 70–82. J. von Neumann, Functional Operators—Vol. II. The Geometry of Orthogonal Spaces, Princeton University Press, Princeton 1950 (a reprint of mimeographed lecture notes first distributed in 1933). J. von Neumann, On rings of operators. Reduction theory, Ann. of Math. (2) 50 (1949), 401–485. C. Niculescu and N. Popa, Elements of Theory of Banach Spaces, Bucarest 1981 (Romanian). ¨ W. Orlicz, Uber konjugierte Exponentenfolgen, Studia Math. 3 (1931), 200–211; Reprinted in: WÃladysÃlaw Orlicz, Collected Papers, PWN, Warszawa 1988, 200–211. W. Orlicz, A note on modular spaces. I, Bull. Acad. Polon. Sci. S´er. Sci. Math. Astronom. Phys. 9 (1961), 157–162; Reprinted in: WÃladysÃlaw

Hidegoro Nakano

169

Orlicz, Collected Papers, PWN, Warszawa 1988, 1142–1147. [Or92] W. Orlicz, Linear Functional Analysis, World Scientific 1992, xvi+246 pp. [Nakano, p. 11, 237]. [PR92] V. Peirats and C. Ruiz, On lp -copies in Musielak-Orlicz sequence spaces, Arch. Math. 58 (1992), no. 2, 164–173. [Pe67] A. L. Peressini, Ordered Topological Vector Spaces, Harper 1967 [Nakano, p. 18, 43, 58, 113, 142, 159, 213]. [Pi07] A. Pietsch, History of Banach Spaces and Linear Operators, Birkh¨auser, Boston 2007, xxiv+855 pp. [Nakano, p. 92, 125, 488, 626, 655, 700, 791]. [Pl71] B. I. Plotkin, General theory of groups, in: Itogi Nauki Ser. Mat., Algebra, Topology, Geometry (1970), Akad. Nauk SSSR, Vsesojuz. Inst. Nauchn. i Tehn. Informacji, Moscow 1971, 5–73; English transl. in J. Soviet Math. 1 (1973), no. 5, 527–570. [PR08] L. P. Poitevin and Y. Raynaud, Ranges of positive contractive projections in Nakano spaces, Indag. Math. 19 (2008), no. 3, 441–464. [RS90] F. Riesz and B. Sz.-Nagy, Functional Analysis, Dover Publications, New York 1990, xii+504 pp. [Nakano, p. 351, 354, 363, 366, 452]. [Sa05] S. Samko, On a progress in the theory of Lebesgue spaces with variable exponent: maximal and singular operators, Integral Transforms Spec. Funct. 16 (2005), no. 5–6, 461–482. [SS03] W. Sanhan and S. Suantai, Some geometric properties of Ces` aro sequence space, Kyungpook Math. J. 43 (2003), no. 2, 191–197. [Sc60] H. H. Schaefer, On the completeness of topological vector lattices, Michigan Math. J. 7 (1960), 303–309. [Sc74] H. H. Schaefer, Banach Lattices and Positive Operators, Springer-Verlag, New York-Heidelberg 1974 [Nakano, p. 78, 80, 138, 139, 295]. [Sc84] H.-U. Schwarz, Banach Lattices and Operators, Teubner Texts in Math., Leipzig 1984. [Se71] Z. Semadeni, Banach Spaces of Continuous Functions. I, Monografie Matematyczne 55, PWN—Polish Scientific Publishers, Warsaw 1971, 584 pp. [Nakano, p. 53, 59, 232, 304, 445, 464, 484, 514, 535]. [Sh67] I. V. Shragin, The Amemiya norm in an Orlicz-Nakano space, Kishinev. Gos. Univ. Uchen. Zap. 91 (1967), 91–102 (Russian). [Sh75] I. V. Shragin, Some metric properties of Orlicz-Nakano spaces, Trudy Mosk. Inst. Khim. Mashin. 64 (1975), 45–52 (Russian). [Sh76] I. V. Shragin, Conditions for the imbedding of classes of sequences, and their consequences, Mat. Zametki 20 (1976), no. 5, 681–692; English transl. in: Math. Notes 20 (1976), no. 5–6, 942–948 (1977). [Sk75] A. I. Skorik, Linear topological classification of the sequence spaces l(pk ), in: Collection of Articles on Applications of Functional Analysis, Voronezh. Tehnolog. Inst., Voronezh 1975, 173–194 (Russian).

170

L. Maligranda

[Sk76] A. I. Skorik, Isometries of ideal coordinate spaces, Uspehi Mat. Nauk 31 (1976), no. 2(188), 229–230 (Russian). [Sk80] A. I. Skorik, Isometries of a class of ideal coordinate spaces, Teor. Funktsi˘ı Funktsional. Anal. i Prilozhen. 34 (1980), 120–131 (Russian). [St49] M. N. Stone, Boundedness properties in function-lattices, Canadian J. Math. 1 (1949), 176–186. [Su99] S. Suantai, On matrix transformations related to Nakano vector-valued sequence spaces, Bull. Calcutta Math. Soc. 91 (1999), no. 3, 221–226. [SS01] C. Sudsukh and S. Suantai, Matrix transformations on the Nakano vector-valued sequence space, Kyungpook Math. J. 41 (2001), no. 1, 35–44. [Su71] K. Sundaresan, Uniform convexity of Banach spaces l({pi }), Studia Math. 39 (1971), 227–231. [Ur85] R. Urba´ nski, A generalization of the Koshi-Shimogaki quasinorm and a topology in the modular spaces in the sense of Nakano, Arch. Math. (Basel) 45 (1985), no. 4, 366–373. [Vo74] Kh. Vo-Khac, Treillis de Freudental et th´eor`eme de Kakutani-Nakano concernant la classe Lp (0 < p < ∞), C. R. Acad. Sci. Paris S´er. A 279 (1974), 495–498. [Vu67] B. Z. Vulikh, Introduction to the Theory of Partially Ordered Spaces, Groningen, Wolters-Noordhoff 1967, xv+387 pp. [Nakano, p. xiii, 144, 245, 382]. [Wa08] D. Waterman, My academic life, in: Topics in Classical Analysis and Applications in Honor of Daniel Waterman, Edited by Laura De Carli, Kazaros Kazarian and Mario Milman, World Scientific Publ., Hackensack, New Jersey 2008, 1–18. [WIBR69] D. Waterman, T. Ito, F. Barber and J. Ratti, Reflexivity and summability: the Nakano l(pi ) spaces, Studia Math. 33 (1969), 141–146. [We60] J. D. Weston, A note on the extension of linear functionals, Amer. Math. Monthly 67 (1960), 444–445. [Wi07] A. Wickstead, An isomorphic version of Nakano’s characterisation of C0 (Σ), Positivity 11 (2007), no. 4, 609–615. [Wi55] N. Wiener, On the factorization of matrices, Comment. Math. Helv. 29 (1955), 97–111. [Wi71] J. H. Williamson, Review of the book by H. Nakano, “Uniform Spaces and Transformations Groups”, Wayne State Univ., Detroit 1968, Bull. London Math. Soc. 3 (1971), no. 3, 377. [Wi58] K. Winkelbauer, Ergodic theorem in complete vector lattices with abstract norm, Czechoslovak Math. J. 8 (1958), 1–21 (Russian). [Wn84] W. Wnuk, Representations of Orlicz lattices, Dissertationes Math. (Rozprawy Mat.) 235 (1984), 62 pp. [Wn91] W. Wnuk, l(pn ) spaces with the Dunford-Pettis property, Comment.

Hidegoro Nakano

171

Math. Prace Mat. 30 (1991), no. 2, 483–489. [Wn93] W. Wnuk, Banach lattices with properties of the Schur type–a survey, Confer. Sem. Mat. Univ. Bari No. 249 (1993), 25 pp. [Wn94] W. Wnuk, An elementary proof of some classical Nakano’s result, Funct. Approx. Comment. Math. 23 (1994), 3–5. [Wn99] W. Wnuk, Banach Lattices with Order Continuous Norms, PWN, Warszawa 1999 [Nakano, p. 9, 75, 103, 121]. [Wn10] W. Wnuk, On significant contribution of Japanese mathematicians to the Riesz spaces theory, April 2010, manuscript, 32 pages. [Wo73] J. Woo, On modular sequence spaces, Studia Math. 48 (1973), 271–289. [Ya09] I. B. Yaacov, Modular functionals and perturbations of Nakano spaces, J. Log. Anal. 1 (2009), Paper 1, 42 pp. [Ya54] S. Yamamuro, Modulared sequence spaces, J. Fac. Sci. Hokkaido Univ. Ser. I. 13 (1954), 1–12. [Ya59] S. Yamamuro, On conjugate spaces of Nakano spaces, Trans. Amer. Math. Soc. 90 (1959), 291–311. [Za83] A. C. Zaanen, Riesz Spaces II, North-Holland, Amsterdam 1983 [Nakano, p. 340, 413, 423, 425, 432, 582, 711–712]. [Za94] R. Zaharopol, On the Nakano individual convergence, Z. Anal. Anwendungen 13 (1994), no. 2, 181–189. [Zl07] B. Zlatanov, Schur property and lp isomorphic copies in Musielak-Orlicz sequence spaces, Bull. Austral. Math. Soc. 75 (2007), no. 2, 193–210. Lech Maligranda Department of Engineering Sciences and Mathematics, Lule˚ a University of Technology, SE-971 87 Lule˚ a, Sweden E-mail : [email protected]