Personal Notes - In My Own Words

When I was eight years old, my elder brother Bengt started to give me math problems, which were different from anything I had seen before, and where there was no indication how they should be solved. For me this was a very important influence. In my mathematical career I have always preferred to work on problems, where the known methods had failed and where one needed to come up with new ideas and methods to solve them.

Other important early influences were my teachers in piano and composition. When I started to do research in mathematics I went about it in the same way as if I should compose a piece of music. I had the idea that I should create my own piece of art. So I made up some systems of equations involving addition and composition of functions, and started to solve them. I showed professor Rådström at Stockholm University what I had done. He suggested that I should study topological groups, which might provide some framework for what I was trying to do. I read the books by Pontrjagin and Montgomery-Zippin on topological groups and that led me to work on an infinite-dimensional version of Hilbert's fifth problem.

For 5 years, 1964-69, I worked in isolation on this project. In this process I developed new insights and methods, which turned out to be useful for solving long-standing, fundamental and famous problems in Functional Analysis.

REMARKS ABOUT EARLY YEARS

When I was seven, my younger brother Hans "forced" my parents to have him start on the piano. So Cecilia, my elder sister, and I also started. After 1-2 months I got passionately interested and begun to play several hours a day. I read about what the great composers did as children, and it was my dream to be like them. Since they composed, I did that too. I read that Grieg had brought his compositions to school and hid them in his school bench. So I did the same. My school teacher let me work on "my piano concerto" - which was little more than random notes - in the "free activity hour". My piano teacher Bengt Utterström would initially only let me study Kuhlau sonatinas etc. and I tried, by myself, to learn Grieg's piano concerto, Beethoven's 3rd piano concerto etc., having the feeling that Utterström, who was really excellent, was holding me back. But I learned more from Utterström than from any of the much more famous teachers I had later.

When I was eight, I discovered mathematics in some form. I was competing with a classmate, who could first get through the book, that we studied. In the process, I got fascinated by this problem solving, and in a couple of months I got through all books including up to the end of 6th grade. Then I was excused from following the math. curriculum, and that was actually the case for many of my later school years. Also, around this time my elder brother Bengt, started to give me math. problems to think about, problems, that had nothing to do with school. I regard this a very important influence. Thinking about some problem, where there was no instruction, how to go about it, created in me the feeling, that mathematics is something, that I should find out by myself how to do - not something that I could really learn from others.

When I was nine, I got interested in sports - especially running. This was the time when Emil Zatopek was at the peak of his running career, and he was my big hero. I read and saw on pictures that Zatopek made painful faces when he was running and I imitated that. For 2-3 years, until we moved from Karlskrona in Southern Sweden to Stockholm, I exercised a lot, often also trying to involve the other children on the street in the sports activities. Although I did not have any great talent, through my school years, I was always one of the best in my class in several sports. To some, this was a surprise, because if somebody did well in school (and especially if that person also liked classical music) he would automatically be bad in sports.

I did not lose the old interests when new interests were added (although there might have been a temporary fluctuation). I had a strong desire to excel in music, math. and sports. I was also very social, I often brought many friends home after school, and then I tried to convince my parents, that it was not really my fault, that they stayed so long. I remember having quite a bit of compassion with those, who were less fortunate than I was - either it was because they needed to spend more time on the school work than I did or because they did not grow up in a good home.

When I was ten, I auditioned for the Austrian piano professor Bruno Seidlhofer. He had become very famous, much through his student Friedrich Gulda, who was the great new piano star in my childhood. Seidlhofer was going to give summer schools in Sweden in 1955 and 1956 (when I was going to be eleven resp. twelve) and to attend I should learn German (which I did on my own through a course on the Radio). Seidlhofer expressed a high opinion of my talents and tried to convince my parents, that I should quit going to school and move to Vienna. He would make me "a new Gulda". My parents were hesitant to do this. My father - although always supportive of my music - believed that I had too much talent for mathematics to be "just a performer". In any case, such a big decision should not be made so early.

At that time, I started to get quite a bit of attention as "the new piano prodigy, who was also good in math. And sports". A music director Erik Vikbladh from the town Åmål asked me to give a piano recital in their concert series. So, in early 1956, when I was eleven, I gave my first piano recital with a full program - Bach, Haydn, Mozart, Beethoven and Schumann. I can only say that it was a tremendous success. In the spring of 1956 a "Mozart competition" (it was the 200th anniversary of his birth) was arranged in Sweden - in piano, violin and wind instruments.

It was for young musicians, not more than 17 years old. There were several hundred participants from all over Sweden. I ended up winning the piano competition. My brother Hans made it to the final in the violin competition although he did not win.

As prize, we got to spend two weeks at Mozarteum in Salzburg and to give a concert there. In the fall of 1956, when I was twelve, the Royal Opera of Sweden arranged a special Mozart program, which was going to be performed 8 times. I was invited to be the soloist in Mozart's 19th piano concerto, which I played 8 times with the Royal Opera Orchestra.

One of the leading Swedish conductors Tor Mann offered me free lessons in orchestral conducting and help to arrange more performances. In early 1956 we had moved from Karlskrona to Stockholm, and I had started to study piano with Gottfrid Boon. He was the leading piano teacher in Sweden at the time and he had just retired from the Royal Conservatory of Stockholm. He was a student of the great A. Schnabel in a very strong sense - he tried to systematize Schnabel's ideas on piano playing and music performance. This had obvious strengths - in terms of developing a good ear for tone quality, shaping of phrases, tight legato playing etc. as well as for the spirit in the works especially by Beethoven, Mozart, Schubert and Brahms. It also had some weaknesses - in terms of not giving enough focus on technical virtuosity, something that certainly was important on the European continent (and Russia). So although I studied some virtuosic literature - some Liszt, almost all Chopin Etudes etc., the main focus was on Beethoven etc. and the technical brilliance, that I had at the age of eleven (combined with solid musical understanding) was not developed like it would have been with Seidlhofer.

The spring of 1956 represented the addition of one more interest (or love): Nature in general and wildflowers in particular. I also read a book about human evolution and was fascinated by it.

I did little or no systematic study of mathematics in these years. That really started only in 1962 when I finished high school. But I was interested in math. problems.

When I was eleven, I saw for the first time a system of linear equations with several unknowns, and after two days I had figured out how to solve such systems. When I was thirteen I figured out how to solve equations of third degree - after having been told how to get rid of the x2 term.

I also tried to prove Fermat's Last Theorem and the twin prime conjecture.

In the years 1957-60, there were more concerts and music studies - now also theory and composition and also orchestral conducting.

Chamber Music at home in Malmö in 1959; Hans is 13 and Per is 15 years old

Three happy musicians in 1960 at the Austrian Embassy in Stockholm receiving "The Austrian Cultural Fellowship" grants for music studies in Austria. Hans Enflo, Per Enflo, and Anders Collden

I received "The Austrian Cultural Fellowship" and used it to study orchestral conducting in Salzburg in the summer of 1960 (with L. von Matacic, L. Hager and as a guest lecturer H. von Karajan).

I discovered poetry in 1958 - especially the poet Friedrich Hölderlin.

In 1958 I read the book Music and Musicians by the great conductor Bruno Walter and that had considerable influence on me.

In the spring of 1961, a national competition for young pianists (up to 20 years old) was arranged in Sweden. There were approx. 300 competitors and I came out the winner. As prize I got to study with the famous pianist Geza Anda in the summer of 1961. At that time he expressed a high opinion of me ("you are very talented") and encouraged me to go full force with a piano career.

In the fall of 1961, for the first time in Sweden, a national mathematics competition for high school students was arranged. I came out the winner of the first round - being the only one of the 550 competitors, who had solved all 6 problems. In the final round I made a mistake (by putting x=a instead of y=a) and although the problem I solved was equivalent to the problem, that was asked for, I got no credit and ended up in second place. But that was not really important.

After doing so well in this math. competition, I felt that it would be wrong, not to continue with some real math. studies.

I have sometimes been asked:

When did you make the decision to go into math. instead of music?

From what I have written, and from what will follow below, my answer to that is quite natural: I have never made, and will never make, any such decision.

Statements like:

"It is, of course, best to have math. as a living and music as a hobby"

such statements I cannot relate to.

In the spring of 1962, I finished high school and I was happy to be able to concentrate on my main interests. I went full force with studying math. And theoretical physics, trying to complete 3 years of studies in one year. I very strongly felt this: If I do not try now to accelerate my piano career, then I better be a mathematician of the highest level. Otherwise I am just wasting my life and the contributions to music, that I am capable of.

I had several fairly big performances in 1962. With different Swedish orchestras I was doing Beethoven's 2nd piano concerto and Franck's Symphonic Variations. Hans and I gave a duo recital in the Göteborg Concert House.

From 1961 on I started to get a new type of requests: To perform at "prestigious" dinners and events - involving business and industry leaders, ambassadors, Swedish nobility and the royal family etc. This continued for 5-6 years, when my popularity started to fade away. One such event in 1962 was arranged by the owners of Örbyhus Slott - an impressive castle, where, in the 16th century, the Swedish king Erik the 14th was held captive.

Many of these requests I got through Joen Lagerberg, a retired diplomat who was "Överceremonimästare" for the Swedish King. I lived in his home during my studies at Stockholm University.

In the spring of 1963, I stumbled in my attempt to complete my 3 years of study, I (barely) flunk the last test, which was also a "qualifying test" for the research program. In September I (barely) flunk it again and started to get worried: After 4 failures one would be black-listed. Anyhow, my third attempt in the early spring of 1964 succeeded.

In the fall of 1963, I was busy preparing my official "Debutkonsert" in the Stockholm Concert House. The program was Brahms' ballades, Beethoven's Waldstein Sonata, Schumann's Fantasie op.17 and Shostakovich preludes. To give a "Debutkonsert" was part of the usual procedure to establish oneself as a musician. There was a lot of talk about the best way to go about choosing program (it was easier to get good reviews on 20th century music than on older music) and also other aspects of this event. My Debutkonsert was successful - the reviews varied from very good to enthusiastic.

In the fall of 1963 I was also preparing my next attempt on the "qualifying test" and following research level seminars. I rediscovered Helly's Theorem on intersections of convex bodies when trying to solve some problem that came up. I was also thinking about how I could make some real contribution to mathematics. I thought, that if I could only find some really good, important project to work on, then I could have a chance of doing it. I did not go to anybody, to ask for a project. As a start, I went about this in the same way that I went about composing music (something that I had also been studying since 1957). I made up some systems of functional equations and started to solve them.

After passing the "qualifying test" in the spring of 1964, I showed to prof. Hans Rådström at Stockholm University, what I had done. He told me, that what I had done could be related to infinite-dimensional topological groups and suggested that I should study some books on topological groups. So I studied Pontrjagin's book and the book by Montgomery-Zippin.

I remember the time until the spring of 1965 as frustrating, sometimes depressing, as far as finding any good direction for my work. I was moving around in the dark, doing some small things, rediscovering some early Banach space results - but not seeing any big picture, not seeing anything that could be a real research project, much less seeing anything that I could perceive as having any importance. And the few talks I had with Rådström did not help me.

As far as my piano career, the picture was mixed. I was back to study with Geza Anda in the summer of 1964. He was disappointed that I had not made more progress since 1961. On the other hand, I gave concerts and in the end of the year I was invited to a prestigious event that had a lot of media attention:

The Christmas Concert in the Stockholm Concert House, arranged by Svenska Dagbladet, one of the big Swedish newspapers - where leading singers, both classical and popular, other musicians, actors etc. performed, each one having 5-10 minutes. I played Schumann's Abegg Variations.

In May 1965, there was some real progress in mathematics. I understood that some groups, that I had been looking at were strange, because the group operations were not locally uniformly continuous - and that this fact could be connected to an infinite-dimensional version of Hilbert's fifth problem. So - suddenly - I had a big project, which seemed important enough, and even a start of some progress on it. I was near to be able to extend to the infinite-dimensional case - by new methods - where the finite-dimensional case had been around 1945, 5 years before the problem was solved.

In the spring of 1965 I was also preparing to take part in Concours Clara Haskil in Switzerland. The competition had run in 1964, but no competitor was found good enough and no winner had been chosen. A huge program had to be prepared, consisting of several recital programs and piano concertos. In the end, only 8 pianists came to the competition. I was one of them, but I had no success. The winner was Christoph Eschenbach. Among the 8, he was in a class of his own, and his victory was an important step in his subsequent world career as pianist (and conductor).

After the first progress in mathematics, I was back to struggling. But it was at a better level than in 1964, and with the many non-working ideas I had, how to go on with Hilbert's fifth problem, I was slowly building up insights on how to handle infinite-dimensional problems. But I was working totally on my own - the only contact I had was Rådström. And I only saw him 1-2 times per semester, to show him what I had done. So, I can here summarize, what I feel were the important influences for my mathematics research. Rådström certainly should be mentioned. He was always supportive of my efforts, and believed that I could do important things. But he could offer little guidance - beyond the suggestion to study topological groups and critizising me, when he thought that my theorems had too complicated statements. What I feel had been more important influences in a deeper sense was Bengt giving me non-standard problems to think about, when I was a child. And my composition teachers and my piano teacher Boon. I was going about doing mathematics research with the idea, that I should create some very personal "piece of art" - in a similar way as composing a piece of music or interpreting a piece of music.

That is an important part of the reason, why I was working in almost total isolation for 5 years, from 1964 to 1969. And that was certainly not the "correct" thing to do. The correct thing, I was told, was to team up with Carleson, Hörmander or Gårding - they were the people with power, who could make or break math. careers. The research program I was pursuing was certainly not in the Swedish tradition of classical analysis. It seemed really quite far from what anybody else in the world was doing or could be interested in.

Together with all my involvement with mathematics, I continued my piano career as well as I could (and that is actually the case until today). One of the concerts in 1966 was particularly nice. A leading English cellist C. Bunting was going to give a recital in Stockholm. He had visited Sweden before and heard me play. He had liked my playing and asked me to do the recital with him. Among other things, we played Brahms' F major sonata. The reviews were excellent - especially for Bunting but also for me.

I made important progress in mathematics in 1966, but it was more on the level of new insights, than actual results. When thinking about topological groups in the spirit of Hilbert's fifth problem (I had gradually modified the Hibert problem to some very general program: To decide whether different classes of topological groups shared properties with Lie groups) I was wondering whether there exist "very non-commutative" groups i.e. groups, where all elements except e, are conjugate to each other. I constructed such groups, by finding the right finite phenomenon and then make an induction. I understood, that this is a very general construction scheme (or "philosophy"), that probably could be applied to various infinite or infinite-dimensional problems. And actually – this philosophy is behind several of my best papers - the solution of the basis and approximation problem, the solution of the invariant subspace problem for Banach spaces, the solution of Smirnov's problem on uniform embeddings into Hilbert space and more.

Other important progress in that year (I got it all together in January 1967) was the introduction of a notion of "non-linear type" in Banach spaces. In 1966 I had understood that to solve the equation x2 = y in certain groups modeled on Banach spaces, one needed some inequality for some geometric configuration. It took me more than half a year to figure out that this configuration is a set, numbered as the corners of an n-dimensional cube. In the 70:s, the notions of type and cotype in Banach spaces became very important and later also the concept of non-linear type. When writing about type and cotype in Banach spaces prof. W. Johnson and prof. G. Pisier wrote (approx.): "Enflo, a little bit ahead of his time, had already introduced a notion of non-linear type ... "

To give an idea of how I related to the "Mathematical Establishment" I should mention a few words about the 1966 summer school of Harmonic Analysis in Sigtuna outside Stockholm. Many "big shots" were there to give lecture series: E. Hewitt, Kahane, Katznelson, W.Rudin, Carleson, and Varopolous. The young elite of classical analysts from the Nordic countries were there. Carleson lectured on his very technical work on almost everywhere convergence of Fourier series. Varopolous was the new, young star. The young elite of analysts seemed to understand everything, and in the evenings they discussed what the big shots had said to them about their ideas. I was totally lost, although I had followed seminars all spring - the organizers had ordered everyone to be prepared for this important event; that is why special seminars had been arranged. Finally, I understood one of the problems that W. Rudin mentioned - I worked on it for a week and I thought that I had a solution. So I was courageous, I went up to him and told that I had solved his problem. As I explained my solution, I realised that I had made a big, stupid mistake. I was very embarrassed but he was nice and said: You play the piano well.

It was soon found out that I played the piano and Hewitt asked me to accompany him, when he played the horn, and I did that. In the final day of the school, he eventually asked me what I was doing in mathematics. So I told him that I was working on an infinite-dimensional version of Hilbert's fifth problem. His reaction was immediate and strong: That cannot be very important. At the end of the school, several of the big shots expressed how happy they were to have given a lecture series at such a high level, and how confident they were, that they could leave the future of the field to the young generation that had attended.

In the spring of 1967 I got my "Fil.lic." degree. It corresponded fairly well to the international PhD, but the Swedish system was different.

In the early summer I had to go to the army for almost one year of military service (military service is mandatory for Swedish young men). My brother Hans had been chosen to study Russian in the army, and he had been able to help me to switch to the same kind of service. But I spent most of my free time on mathematics and music. By using my "philosophy" I showed that local groups cannot always be extended to full groups, one of the problems in Pontrjagin's book. The problem was in the second edition of the book, I had checked that it was not solved after the second edition, only later I found out that it had been solved in the time between the first and second edition. But in all the things I was trying to do on topological groups, it often happened that somebody had already done what I was thinking of.

Around Christmas 1967, I thought, that I should try to apply my philosophy to the Basis problem. At that time it was - together with the Approximation problem - the most important and famous open problem in the theory of Banach spaces. I played around with it for a couple of weeks. I did not see any finite phenomenon, that could be used for some construction. But I still made some progress, I realised that a basis must have a "basis constant" and I understood why some of the natural ways to construct bases do not work in general.

In the spring of 1968, Hans and I had a one week concert tour in Southern Sweden, playing recitals every day. We had been hired by Rikskonserter, a government agency for providing all of Sweden with music. We also gave a concert in Stockholm with the same program. It was a successful tour and we got good or very good reviews. My military service was coming to an end, and they were nice to me and let me have one week off. In the summer we prepared to take part in an international duo competition (violin-piano) in Munich. We had to prepare 8 sonatas - among them Beethoven's Kreutzer and Franck's A major sonata. We were fairly well prepared but had no success.

In the summer of 1968, I was able to show that Lp-spaces and Lq-spaces are not uniformly homeomorphic if p and q are different, by using my "non-linear type" in simple way. My proof worked for p and q between 1 and 2. I was told by S. Schonbeck at Stockholm's University, that there was somebody named Lindenstrauss, who had done something, that probably was related. I checked Lindenstrauss' paper and saw that his result precisely complemented mine - he had been able to cover those p and q not covered by my method. In his paper there were also some other problems: Smirnov's problem whether all separable metric spaces are uniformly homeomorphic to subsets of Hilbert space and the problem whether Banach spaces can be uniformly homeomorphic without being linearly isomorphic. Smirnov's problem had been around for approx. 15 years, so it was an inspiring challenge. By applying my philosophy and a stronger notion than non-linear type, I was able to solve the problem in a little more than 2 months. Rådström was impressed and felt that this should be published in Acta Math., even if it was a short paper. So I sent it to Carleson for publication in Acta. He immediately rejected it - with criticism that seemed irrelevant to me. Rådström told me not to say anything, because "Carleson has power and enjoys using it". I followed the advice and the paper was published in Arkiv.

The success with Smirnov's problem was an inspiration to come back to the Basis problem and I did so in January of 1969. After a couple of months, I found that there was a finite-dimensional phenomenon, upon which an induction could be applied, to construct a counterexample. But this was apparently very complicated, one needed to control projections on a huge number of spaces, and it looked impossible to do so. In the late spring, I understood, that I could test, if my philosophy could work, by shooting for a weaker counter-example - a Banach space, for which there is a number a > 1 such that every basis has basis constant > a. By doing so, I could work just in low-dimensional spaces. Still, I would get a strong indication whether all separable Banach spaces have bases or not.

In December of 1969, I managed to put it all together, so I had the strongest result on the Basis problem. My philosophy had worked, I was convinced that there are Banach spaces without Schauder basis and I "knew in principle" how to construct them. But the technical difficulties seemed formidable.

There was other progress in 1969. In the summer I was able to prove, that if a Banach space is uniformly homeomorphic to Hilbert space, then it is linearly isomorphic to the Hilbert space. This result did not use my philosophy, I got it in other ways. The result was very much in the line of Hilbert's fifth problem, but it was also a result in Banach space geometry, that others had tried to prove or disprove. In a recent book by Deville, Godefroy and Zizler it is referred to as "one of the pioneering results of Enflo in this area".

Another important and exciting thing happened in September 1969: My isolation was broken. The two leading functional analysts in Israel and Poland, Lindenstrauss and Pelczynski, visited Stockholm for a couple of days. They did not know about my existence and were very surprised. After I told them of some of my results, Pelczynski asked me: Do you know, that there is a problem, whether a Banach space that is uniformly¨homeomorphic to Hilbert space must be isomorphic to Hilbert space? So I said: Yes, I have proved that. I presented the solution in a hastily arranged seminar. Both Lindenstrauss and Pelczynski expressed strong enthusiasm for several of my results.

After I met Lindenstrauss and Pelczynski I stopped working on more general topological groups and turned to Banach spaces (which, of course, are special cases of topological groups). My earliest work received little attention for many years, but that may slowly change. In the recent book on non-linear functional analysis by Benyamini and Lindenstrauss, there is one chapter which contains just part of my earliest work. Lindenstrauss told me in 1996, that when they were writing the book, he had for the first time gone through this work carefully, and had been impressed. My earliest work contains beginnings of several directions of research, directions, that are still unexplored. But the notions of non-linear type, that I introduced, have now also become important in other areas of Mathematics and in Computer Science.

In 1969 Rikskonserter offered me another tour. It was to accompany a singer B. Hedeby in 22 concerts, most of them in schools in Southern Sweden. The whole tour took two and a half weeks (in October), it was a lot of hectic driving to get on time to the different events. My cooperation with Hedeby continued until 1971, when I moved to Berkeley. It took me to give performances in places that I had seen little of before: Hospitals for long-term treatments, drug rehabilitation centers, prison, meeting places for the Social Democratic Establishment etc.

In April 1970 I defended my thesis for "Filosofiska Graden". This degree was part of the old Swedish system, it had been cancelled a couple of years before, but in a transition period one could still pass it (so I was one of the last to get it). The defense was a public event, there were three "opponents" who would ask questions and make comments, and then everybody was allowed to be an "outside opponent". The first opponent was (formally) chosen by the university, even if, in reality, the "defendant" could usually make the choice. The second and third opponent were chosen by the "defendant". The first and second opponent were serious, the third opponent was traditionally somebody, who did not understand anything of the subject and was making jokes and asking silly questions. My mother had convinced me to arrange that Lindenstrauss would be the first opponent. She thought that it would be good for my career in Sweden to have somebody of international reputation, who could appreciate my work. This was a good idea but - for anyone who had any understanding of the politics in Swedish mathematical life - it was also a naive idea. The second opponent was Edgar Asplund. Asplund was a very good, internationally known, Swedish functional analyst, who had spent much time abroad. He was never accepted by the Establishment in Swedish mathematics and could never get a professorship. He applied several times but was usually formally declared "incompetent". He became a professor in Denmark, where he died, too early, from cancer in 1974. I had little contact with him before 1969, but after that we kept in contact until he died. My third opponent was Joen Lagerberg, who had never studied any mathematics.

After my thesis defense I applied for positions. There were several openings for "docent" positions (a common step in the old Swedish career) in 1970 and 1971 and I applied for them. Rådström had said to me:" If there is any justice in the world, you should have one of these positions". Of the applicants I had the best grade on my thesis for Filosofiska Graden. And I had my result on the Basis problem and some other work. But I never came in better than in third place. An applicant, who had defended his thesis half a year after me and obtained a lower grade, was considered more qualified because the work he had done outside his thesis was "of even greater value". In one case, a professor Roos called me up before the evaluation and told me that my result on the Basis problem was not new, Gurariy had already done it. I tried to explain to him that my result was (much) stronger than Gurariy's. Still, in the evaluation report, there was no positive word about my result, only criticism for lack of comparison to Gurariy's result. A professor Lech at Stockholm University told me, that my way of working on long-standing problems represented social ambitions rather than mathematical ambitions. Professor Jaak Peetre in Lund had a good opinion of my work. He tried to convince Lars Gårding to open up a "docent" position for me at Lund University, because Gårding had the power to do that. But Gårding told him that there was no reason to make any action for me.(However, later, Gårding became very supportive of me, at least for some time. In his popular book about mathematics "Encounter with Mathematics", he mentions only two Swedish mathematicians, Fredholm and me).

If things looked difficult in Sweden, they looked better outside. In November 1970 I was nominated for a "Miller Research Fellowship" at University of California at Berkeley and in the spring of 1971 I got it.

Rådström died in the fall of 1970, just before my nomination for Miller Fellowship. He had been a heavy smoker and had a heart attack. With his professional integrity, honesty and openmindedness he was a good force in the mathematical life in Sweden. For me personally it was a big loss, he had always been very supportive.

I continued to work on the basis problem in 1970, in a somewhat new direction. Lindenstrauss was at that time convinced that the "Approximation problem" - a somewhat more general problem than the Basis problem - had a positive solution for uniformly convex Banach spaces, and he had an approach. I tried this approach without success. In doing so I came up with the concept "minimal point", a special case of an old concept in Economics, Pareto optimal point. This turned out to be fruitful or different types of non-linear problems, and later had its own development. In looking at the Approximation problem instead of the Basis problem, I saw that my "philosophy" would lead to slightly different finite-dimensional problems than the Basis problem. In some sense, they were more difficult, because they were more general - in another sense they were easier, because they were more "symmetric".

1970 was a Beethoven year (200th anniversary of his birth) and in the spring a series of concerts with all his piano sonatas was arranged in Stockholm. The pianists had all been Boon students. So in May I gave a recital with the Hammerklavier sonata and the op.111, two very demanding works. I got mixed reviews - good in one paper and bad in the other.

In 1970 I got my first doctoral student, Lars-Erik Andersson. He had been Rådström's student for PhD (in the new system, that had been introduced in Sweden, to adjust to the international system) and when Rådström died, I became the advisor. Andersson and I both worked hard through the summer of 1971 so he could finish his thesis work just before I moved to Berkeley.

In the end of August 1971 I came to Berkeley, and I was immediately happy there. There had been unpleasant labor conflicts in Sweden in the spring of 1971, involving university employees. Also my career situation in Sweden was not good.

Prof. Haskell Rosenthal was my "host" and it was very inspiring to start to work, both on my own and with him. It started very well, and in the first month I proved that a Banach space has an equivalent uniformly convex norm if and only if it has an equivalent uniformly smooth norm. That answered a 30 year old, quite famous, question in Banach space theory. R.James had earlier made progress on the problem and solved "half" of it, but I got immediately much attention for my contribution. When working in the stimulating environment in Berkeley, I could combine new impulses with all the insights I had built up in my isolation in Sweden. There was progress on several good projects. In early February of 1972, after a sleepless night, I decided to come back with a big, new push on the Basis problem, or rather the Approximation problem. I had obtained a much better feeling for handling complicated finite-dimensional problems and I felt that I could have a real chance to solve it with my "philosophy". I thought, that even if it would take me several years, it would be worth it. When I started, things seemed to fall into place better than I had expected. But the way I needed to use symmetries of some finite-dimensional spaces made it hard for any induction procedure to work.

Around April 10 I made decisive progress. I got the idea (I do not know how) that one can use "jump in average trace" to estimate norms of operators on finite-dimensional spaces with enough symmetry.

April 1972 was also in other ways an intense period. I had gotten married in 1971 and my first daughter Karin (born 3/29) was just a couple of weeks old. And I was up changing diapers almost every night. The idea of "average trace" made an induction procedure much easier and felt that I had a formidable tool to complete a construction. There were still some ideas missing. In the end of April I almost had an induction working ("if only the harmonic series would be convergent"). And around May 10, when I was out walking with Karin in the stroller, I got the final missing idea. After that it was just to work out details to complete the solution.

The success led to several changes. Carleson wrote a letter to me, asking me to publish the paper in Acta Math. Also Hörmander strongly suggested that I should publish in Acta. And, following Rådström's old but, in this case, naive advice to "sit in the boat" , I did that . In the year 1972-73 I got offers from 6 universities with all possible ranks: Full professor with tenure (Ohio State and Univ. of Utah), 5 year Assoc. prof. without tenure (Stanford and Caltech), 2 year Assoc. prof. with promise of continuation (Univ. of Michigan), highest level Asst. Prof. with a written promise of tenure within a year (UC Berkeley). Gårding asked me if I would still be interested in a "docent" position at Lund Univ., so I told him: "I am sorry; my market value has become too high".

I was invited to Warsaw and visited there in December 1972. Before the visit, I was told that I would receive an unusual prize there. It turned out to be a live goose, offered by prof. Mazur, one of the few mathematicians from Banach's time, who was still alive. Mazur had offered this prize in 1936 for a problem, that he put in the "Scottish book". It turned out in the 50:s, through Grothendieck's work, that Mazur's problem was, in fact, equivalent to the Approximation problem (Mazur when posing his problem, knew only of an implication in one direction). The Basis problem and Approximation problem were discussed in the 20: s the Approximation problem is probably the oldest.

Interdisciplinary Science

Through all my student years, I was interested in interdisciplinary science. There were several backgrounds for that, I will mention 3 of them. I start with the math.-music background. The fact that I had reached a quite high level in two different areas made me often think in what way it could be valuable. I sometimes thought: Perhaps it is just a waste of time and energy to pursue two so different directions. And I sometimes thought: It is too bad, that there is no discipline such as "math.-music", since I could have done a great contribution in such a discipline. And, more importantly, I sometimes thought: I have worked on quite different types of problems: artistic problems, combining emotions, intuition and intellect, and math. problems, combining intuition and intellect. Perhaps that has given me good training to work on problems that have some mathematical nature but are not precisely mathematics. An example: I admire the discovery of the C14 method for dating old material. It is a simple idea of a mathematical nature, but an enormously important idea.

Another background was my other interests, i.e. my interest in wildflowers. I read books about plant physiology, plant geography, ecology and climate. In these areas, there were obviously many fascinating problems like: Why was there a big climate change in Scandinavia around 500 B.C. when the Bronze Age turned into the Iron Age?

For me there was only a small start in interdisciplinary work before 1978-79. In 1963, my brother Bengt was working on some aspects of the Schrödinger equation and for a short time we worked together. And in 1970 I contacted the Royal Inst. of Tech. in Stockholm to do some experiments on music performance. I got some curves on how different pianists performed the beginning of the slow movement of Beethoven's sonata op. 10 no 3. It was intended as a start of a bigger study, but the work never continued after I moved to Berkeley.

Per Enflo was born on May 20, 1944, in Stockholm, Sweden. His father was a surveyor, his mother an actress. Per Enflo is one of five children born to his parents. His family has been, and is, very active in music and other performing arts, and this involvement has been a strong influence in his life.

During his school years, the family moved to various places in Sweden, but Enflo enjoyed a stable, happy home life and good schooling. Around the age of eight he became interested in both mathematics and music. These are the two subjects that he was prodigious in and to which he remains most devoted. You are reading about him because of his mathematics, but in fact, Enflo is almost equally a musician and a mathematician. In music, Enflo has studied piano, composition, and conducting. His first recital was given at age eleven.

In 1956 and 1961 he was winner of the Swedish competitions for young pianists. (...)

Though devoted to both mathematics and music, it is the former that has determined where Enflo has lived. All of his academic degrees have been awarded by the University of Stockholm. Since completing his education in 1970, Enflo has held positions at the University of Stockholm, the University of California at Berkeley, Stanford University, the ´Ecole Polytechnique in Paris, the Mittag-Leffler Institute and Royal Institute of Technology in Stockholm, and at the Ohio State University. Since 1989 he has held the prestigious position of “University Professor” at Kent State University.

Per Enflo is most well known for his solutions, in the 1970s, of the “approximation problem”, the “basis problem”, and the “invariant subspace problem.” These were three fundamental and famous problems from the early days of functional analysis. Since the 1930s, many mathematicians had tried to solve them, but they remained open for about 40 years. The solutions are negative in the sense that they solved by counterexamples; they are positive in the sense that the new methods and concepts have had a great impact on the further development of functional analysis. (...)

Per Enflo’s solution to the approximation problem also gives a counterexample to the basis problem. This work was started in 1967 and completed in 1972 and is a long story of progress and failures and of slowly developing new insights and techniques for final success.

Arguably, his most famous mathematical contribution thus far is his solution to the invariant subspace problem. He constructed a Banach space X and a bounded linear operator T : X -> X with no non-trivial invariant subspaces. The paper containing this example was published in 1987, but it had existed in manuscript form for about twelve years prior to that date. The published paper is 100 pages long, and contains very difficult mathematics. His work on the invariant subspace problem was accomplished during the years 1970–1975, so one can see that the late 1960s to the mid 1970s was a period of remarkable brilliance for Enflo. Enflo’s counterexample, though it gives a complete solution to the invariant subspace problem, left open many doors for future research. For example, determining classes of operators that must have invariant subspaces (in the spirit of Lomonosov’s result) is an active area, and, equally, some of the mathematics developed in his solution to the invariant subspace problem have led him to progress in other areas of operator theory.

The mathematical work discussed in the last few paragraphs might seem particularly abstract, but parts of the associated work have genuine applications. For example, some of the best available software algorithms for polynomial factorizations are based on ideas found in Enflo’s solution to the invariant subspace problem. Also, there are indications that his Banach space work might have good applications to economics.

Enflo’s other important mathematical contributions include several results on general Banach space theory, and also his work on an infinite-dimensional version of Hilbert’s 5th problem.

Enflo’s early career as a musician is an important background for both his originality as a mathematician and for his strong interest in interdisciplinary science. He has done work in biology, on the zebra mussel invasion and phosphorus loading of Lake Erie (work funded by the Lake Erie Protection Fund). In anthropology he has worked on human evolution and has developed a “dynamic” population genetics model that lends strong support for a multiregional theory of human evolution. He has also published work in acoustics, on problems related to noise reduction.”

Extract from the book

Beginning Functional Analysis, Undergraduate Texts in Mathematics,

Springer-Verlag New York 2002 (p. 122-124) by Karen Saxe

Reproduced by kind permission from the author

Updated July 8, 2020 www.perenflo.com © 2011