# Hidegorô Nakano

### Quick Info

Tokyo, Japan

Detroit, Michigan, USA

**Hidegorô Nakano**was a Japanese mathematician who is remembered today for Nakano spaces, and for his results on vector lattices and operator theory in Hilbert spaces.

### Biography

**Hidegorô Nakano**'s parents were Katsugoro and Kame Nakano. Katsugoro Nakano's ancestors had a lacquerware business making ornamental parts for swords. Katsugoro had been brought up to enter the family business and to eventually take charge of it but when the market for such things became less he moved into the business of painting commercial signs. Katsugoro and Kame Nakano had four children, two boys and two girls; Hidegorô was the eldest of the boys.

Hidegorô attended schools in Tokyo, completing his secondary education at a middle school. He then entered the National First High School which was a prestigious preparatory school first founded in 1874 as the Tokyo School of English. Later, in 1886, it moved to the Hongo district of Tokyo. Entrance to the school was very competitive and graduates were almost assured entry into Tokyo Imperial University. The level of education at this school was roughly that of the first two years of a modern UK university. Nakano then entered the Imperial University of Tokyo where he studied mathematics. He was advised by Takuji Yoshie (1874-1947) who had been a student of Lazarus Fuchs, obtaining his doctorate in 1899 with a thesis on differential equations. Nakano graduated from the Imperial University of Tokyo in March 1933 with a Bachelor of Science degree and he continued studying at the Graduate School for his doctorate.

Nakano entered Graduate School on 1 April 1933 but at this time he already had six papers published, all written in German:

*Über die Verteilung der Peanoschen Punkte einer Differentialgleichung*Ⓣ was communicated by Tatsujiro Shimizu to the Physico-Mathematical Society of Japan on 30 October 1931;

*Über den Konvergenzradius der Lösung einer Differentialgleichung $\large\frac{dy}{dx}\normalsize = f(x, y)$*Ⓣ was communicated by Takuji Yoshie to the Imperial Academy of Japan on 12 February 1932;

*Über den Konvergenzradius der Lösungen eines Differentialgleichungssystems*Ⓣ was communicated by Takuji Yoshie to the Imperial Academy of Japan on 12 April 1932,

*Über den Konvergenzbereich einer zweifachen Potenzreihe und seine Anwendungen*Ⓣ was submitted to the

*Japanese Journal of Mathematics*on 13 May 1932;

*Über eine stetige Matrixfunktion*Ⓣ was communicated by Teiji Takagi to the Imperial Academy of Japan on 13 June 1932. Nakano's remaining 1932 paper was communicated by Takuji Yoshie to the Imperial Academy of Japan on 12 October 1932.

Having six papers in print before entering graduate school, Nakano submitted a thesis for his doctorate the day he entered on 1 April 1933. This was even more remarkable than it might at first appear since at that time doctorates in Japan were normally awarded to those over 50 years old. Kentaro Yano, explained the circumstances in [3]:-

On the day that Hidegorô Nakano became a research student, he went to his supervisor's office and applied for doctor degree by presenting his thesis which was already completed. Let me explain now the situation at that time. It was, of course, true that the doctorate was awarded only after a thesis had been submitted, examined and approved. But at that time it was common that a person was allowed to present their thesis only after they had been doing research for about 20 or 30 years and had reached a suitable age. But Nakano had just graduated from the university, and presented his thesis for a doctorate on the same day that he entered graduate school. This also astonished his supervisor, Professor Takuji Yoshie. After a while Nakano told me why he did so. The reason was that there was no age limit for presenting a doctoral thesis and, according to the regulations, there was no examination fee charged if a person was a student in the graduate school. But of course this was indeed a special case. Therefore, Professor Takuji Yoshie examined Nakano's thesis very carefully. He arrived at the conclusion that the author of the thesis was well qualified to be awarded the title of doctor. In 1936, Nakano was awarded doctor of science degree three years after submitting. At that year, he was 27 years old. This 27 year old doctor of science was reported many times in newspapers. The reason was that the people who got a doctorate were normally at least over 50 years of age at that time.Nakano undertook research at the Graduate School until 29 April 1935, when he became a professor at the National First High School. On 24 October 1935, Nakano married Sumiko Yamamura (11 December 1913, Tokyo - 5 March 1999, Detroit). Sumiko was the daughter of Toyonari Yamamura and Akiko Maeda. Hidegorô and Sumiko Nakano had two children both born in Tokyo: a daughter Kazumi Nakano (born 21 June 1937) and a son Hideaki Nakano (born 7 April 1943). He left the position at the National First High School in April 1938 and undertook research at home until he was appointed as a research assistant at the Imperial University of Tokyo on 30 September 1939. On 31 March 1940 he was made an Instructor, a position he held for two years. It was during this period that Japan entered World War II, signing a pact with Germany and Italy on 27 September 1940, but hostilities did not begin until Japan attacked the United States fleet in Pearl Harbor on 7 December 1941. At first the war had little effect on those living in Japan and life continued much as normal. Nakano was promoted to assistant professor on 15 April 1942.

From June 1944 Japan became the target of air raids, initially on military targets. However in February 1945 Tokyo became the target of firebombing raids and quickly the Imperial University of Tokyo was closed with students and staff being evacuated. Kazumi Nakano, Hidegorô Nakano's daughter, wrote in a letter of 18 October 2010 about how her father helped students during the last year of World War II (see [2]):-

Nakano was helping with food for students, but this 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 in the village of Osachi in the Suwa region of Nagano prefecture. Shortage of food was everywhere. Nakano often led his students on a hike hunting for food in neighbouring 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 family maintained a temporary residence and fed hungry students.Nakano's first book was

*From Riemann integral to Lebesgue integral*(Japanese) published in 1940. Next came

*Allgemeine Spektraltheorie:*1940-41 , a collection of his papers written in German followed by three books in Japanese,

*Hilbert Space Theory*(1946),

*Classical Integration Theory*(1949), and

*Measure Theory*(1950). He then published four books written in English:

*Modern Spectral Theory*(1950);

*Modulared Semi-Ordered Linear Spaces*(1950);

*Topology and Linear Topological Spaces*(1951); and

*Spectral Theory in the Hilbert Space*(1953).

More information about these English books can be seen at THIS LINK.

By the time that

*Spectral Theory in the Hilbert Space*was published, Nakano had left Tokyo and moved to Sapporo to take up a professorship at Hokkaido University. He took up this new position on 1 April 1952. Kazumi Nakano, Hidegorô Nakano's daughter, writes:-

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 programme was more like doctoral candidacy today. Regular members of Nakano's seminars, undergraduate as well as graduate, called themselves Nakano's students. Nakano moved to Hokkaido University from Tokyo University in 1952. Both universities were part of the system of the seven 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 appointment at Hokkaido University was thus a move from Tokyo to Hokkaido.Kazumi Nakano also wrote about Hidegorô Nakano's seminar at Hokkaido University:-

The Nakano Seminar existed from 1951 to 1959 while Hidegorô 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 on most of them but some left the university without completing the degree requirements.In 1960 he went to Queen's University in Canada for a year to work with Israel Halperin (1911-2007). Halperin had been a student of John von Neumann and spent most of his career at Queen's University. He had severe problems in the 1940s when arrested and accused of espionage but after a trial in 1947 he was cleared of all charges. Before this visit, Halperin and Nakano had written two joint papers,

*Discrete semi-ordered linear spaces*(1951) and

*Generalized $l^{p}$ spaces and the Schur property*(1953). Nakano wrote in April 1960:-

I am going to Canada, being invited by Canadian Mathematical Congress. I am scheduled to leave Sapporo 26 May, and to arrive at Vancouver 23 June. I shall stay in Kingston for about a year.From February 1961 to March 1974 Nakano was a professor of mathematics at Wayne State University, Detroit, USA. In 1962 he attended the International Congress of Mathematicians in held in Stockholm, Sweden, from 15 August to 22 August 1962. He returned to the United States flying via London to New York arriving on 2 September 1962. He gives the address to which he is going as Wayne State University, Detroit 2, Michigan and his permanent address as No393, 4-chome, Ninami-chinagawa, Chinagawa-ku, Tokyo. Nakano's two children came to the United States to join their parents in 1962.

At Wayne State University, Nakano supervised the studies of eight students for a Ph.D. He continued to work there until 1973 when he was taken ill. His daughter Kazumi Nakano wrote:-

Hidegorô Nakano was suddenly hospitalised in Detroit on 27 August 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 11 March 1974.From the late 1960s Nakano became interested in set theory and attempted to produce a different version of the theory from the classical axiomatic approach. He tried to publish a book giving his approach to set theory but was unable to get it published. He resorted to giving lectures at meetings of the American Mathematical Society and the abstracts of these were published. For example

*Mathematical set theory*(1966),

*On the continuum problem*(1967),

*On the existence of regular initial numbers*(1968),

*Critical numbers*(1968),

*Elimination of the paradoxes in the set theory*(1969),

*An axiomatic set theory*(1969), P

*ointless axiomatic set theory*(1969), and

*The Russell's paradox in set spaces (Axiomatic set theory is not set theory)*(1969). Nakano explained in 1969:-

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 unfavourable 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.Kentaro Yano wrote [4]:-

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-years 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 Christmas 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."Although it is claimed Nakano's

*Set Theory*book was published in 1978, we have not been able to find any trace of this book.

Paul Halmos [1] wrote:-

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.Let us end with saying a little about Kazumi Nakano, Hidegorô Nakano's daughter. She was educated in Japan, studying at Fuji-Gakuen High School (graduated March 1956), Hokkaido University (graduated with a B.S. in March 1960), and then the Graduate School of Hokkaido University (graduated with an M.S. in March 1962). After the award of her Master's Degree, she came to the United States in 1962 and studied for a doctorate in mathematics at Wayne State University. She was advised by Bertram J Eisenstadt (1923-2003) and was awarded a Ph.D. for her thesis

*Uniformity and real Valued Functions*(1967). She worked at Wayne State University, was a member of the American Mathematical Society for 51 years, and died in Brockport, New York, USA on 4 February 2017.

### References (show)

- P Halmos, Hidegorô Nakano, in
*I have a photographic memory*(American Mathematical Society, 1999), 257. - L Maligranda, Hidegorô Nakano (1909-1974) - on the centenary of his birth,
*Proceedings of the International Symposium on Banach and Function Spaces, Kitakyushu, Japan, 2009***III**(2011), 99-171. - K Yano, Hidegorô Nakano (1909-1974) (Japanese), in
*Strange Mathematicians*(Shincho Publishing Company, Tokyo, 1984), 125-134.

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Written by J J O'Connor and E F Robertson

Last Update March 2021

Last Update March 2021