Joram Lindenstrauss, inMemoriamCompiled by William B. Johnson and Gideon Schechtman

Joram Lindenstrauss was born on October28, 1936, in Tel Aviv. He was the onlychild of parents who were both lawyers.Joram began his studies in mathematicsat The Hebrew University of Jerusalem in1954 and completed his PhD in 1962 under

A. Dvoretzky and B. Grunbaum. After postdocs atYale University and the University of Washington,he returned to the Hebrew University, where heremained until retiring in 2005. He passed awayon April 29, 2012.

Joram met his wife, Naomi, during his studies atthe Hebrew University. Naomi holds a PhD degreein computer science from Texas A&M University.They have four children, all of whom have PhDs:Ayelet and Elon are mathematicians at IndianaUniversity and The Hebrew University of Jerusalem,respectively; Kinneret Keren is a biophysicist atThe Technion, and Gallia is a researcher at theInstitute for National Security Studies at Tel AvivUniversity.

Joram’s PhD dealt with extensions of linearoperators between Banach spaces, leading also tothe study of preduals of L1 spaces. Some of hisother groundbreaking research results include astudy with A. Pełczynski of Grothendieck’s workin Banach space theory and applications thereof,which also led to the introduction of Lp spacesand their study, a topic which he continued topursue with H. P. Rosenthal. Lindenstrauss andPełczynski promoted in their paper the “localtheory of Banach spaces,” which involves the studyof numerical parameters associated with finite-dimensional subspaces of a Banach space and the

William B. Johnson is A. G. and M. E. Owen Chair and Dis-tinguished Professor at Texas A&M University. His emailaddress is johnson@math.tamu.edu.

Personal and family photos of J. Lindenstrauss providedcourtesy of the Lindenstrauss Family Album. Used withpermission.

DOI: http://dx.doi.org/10.1090/noti1205

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asymptotics of the parameters as the dimensionsof the subspaces tend to infinity. With L. Tzafririhe solved the “complemented subspace problem,”showing that, isomorphically, the only Banachspaces all of whose subspaces are complementedare Hilbert spaces. The proof uses Dvoretzky’stheorem on Euclidean sections of convex bodies,a topic Joram returned to in an influential paperwith T. Figiel and V. Milman and a related one withJ. Bourgain and Milman.

26 Notices of the AMS Volume 62, Number 1

Early in his career, in a paper on nonlinearprojections in Banach spaces, Joram introducedthe study of nonlinear Lipschitz and uniformequivalences between Banach spaces. Surprisingly,it turns out that often nonlinear bi-Lipschitz or evenbiuniformly continuous nonlinear maps preservethe linear structure to some degree. This topicappears on and off during his career, and hislast publication, a research book with D. Preissand J. Tišer, deals with Lipschitz functions onBanach spaces. The two of us were also attractedto this topic and had the good fortune to cooperatewith Joram on it (sometimes together with others,notably Preiss).

In a paper of Johnson and Lindenstrauss thereis a relatively simple lemma which is widely used,mostly in connection with theoretical computerscience, and which is by far the most quotedresult of Joram. It states that n points in Euclideanspace can be mapped into an approximately logndimensional Euclidean space while approximatelypreserving the pairwise distances. Curiously, butnot coincidentally, the paper does not appear in thelist of selected publications that Joram preparedin his last year.

Joram Lindenstrauss wrote several very influ-ential books. His two-volume book with Tzafriri,Classical Banach Spaces, as well as his book withY. Benyamini, Geometric Nonlinear Functional Anal-ysis, is a must read for anybody interested in thelinear (respectively, nonlinear) theory of Banachspaces. With Johnson, Joram edited the two-volumeHandbook of the Geometry of Banach Spaces.

Joram had twelve official PhD students. Allbut two of them hold/held respectable academicpositions, most in Israel.

Joram’s many honors include the Israel Prizefor Mathematics and the Banach Medal. He wasa member of the Israel Academy of Sciences andwas a Foreign Member of the Austrian Academy ofSciences.

Yoav BenyaminiIn 1965 Joram Lindenstrauss joined the HebrewUniversity as a new faculty, and I was a third-yearundergraduate student sitting in his Banach Spacescourse. This first encounter set the course of mylife.

The Banach space group in Jerusalem beganwith Dvoretzky, who was joined by his formerstudent Grünbaum and their students. A phasetransition occurred with the arrival of Joram, whojoined Dvoretzky in the supervision of the doctoraltheses of Lazar and Zippin, and Joram’s influence

Yoav Benyamini is professor emeritus at Technion-IsraelInstitute of Technology. His email address is yoavb@tx.technion.ac.il.

dominated their work. Soon Gordon started to workwith him on his PhD; I started to work with himon a master’s thesis in 1966. Joram gave excellentbasic and advanced courses, and within a few yearsthe list of his students grew very fast, with Aharoni,Arazy, Schechtman, and Sternfeld. New facultyPerles, Tzafriri, and Zippin joined the department.We had a very active weekly seminar, many visitors,and intensive informal discussions. Soon Joramand Tzafriri wrote their Springer Lecture Notes andthen the two-volume “Classical Banach Spaces”. Byreading proofs of the lecture notes and books,students knew in real time what was happeningthroughout Banach space theory.

Joram’s supervision style was very “open.” Wedid not have orderly weekly meetings, and he nevergave me a problem for the thesis. The exposureto the different directions and problems camethrough his comments and criticism in the seminarand other discussions. This is also how I learnedhow to judge what is “important,” “interesting,”what is worthwhile to read, and what is publishable.His approach and views were so dominant that hewas “heard” even when he was not present.

Joram was very systematic and methodical. Hisanswers to questions sounded like he had prepareda lecture on the subject. He did not like to discussspeculations: he would send me to write up myideas and, of course, in most cases the speculationled nowhere. But when I did hand him somethingin writing, he was a wonderful reader. It wouldcome back the next day with detailed feedback. Hewas also a wonderful writer, and his papers, books,and lecture notes (in Hebrew) are written in histypical systematic, clear, and concise style.

It was a pleasure to work on our book GeometricNonlinear Functional Analysis and to benefit fromhis excellent judgment, his clear view of the bigpicture, together with his care for details, andhis clear and careful writing. Joram was quitedisappointed when I was not very enthusiasticabout writing the planned second volume. Thetwo main subjects were supposed to be the theoryof finite metric spaces and differentiabililty ofLipschitz functions on Banach spaces. I thoughtthat they were not ripe yet for a book. Joram didthe right thing and cooperated with two experts,Preiss and Tišer, to write their major researchmonograph Differentiability of Lipschitz Functionsand Porous Sets in Banach Spaces, which will bethe basis for any future study of the difficult andimportant topic of Fréchet differentiability. Ourlast conversation was when I called to congratulatehim on Elon’s Fields Medal, and he told me proudlythat the manuscript of the book was just sent tothe publisher.

Most of Joram’s students remained in academia,mostly in Israel. Israeli mathematics was very

January 2015 Notices of the AMS 27

important to him. He stopped taking new studentsbecause Israel is a small country that could notabsorb more Banach space experts. As editor ofIsrael Journal of Mathematics his commitment tohigh standards was also influenced, as he toldme several times, by the high quality we haveto preserve of anything that carries the brandname of Israel. He was an active representative,on behalf of the Israeli Academy of Sciences, inthe IMU and talked seriously about applying forthe organization of the International Congress inJerusalem before the murder of Prime MinisterRabin and the increased terrorist attacks put anend to this dream.

Joram set my professional career, and I tried tofollow the values he installed in me. I was also verylucky to be a student of his devoted wife, Naomi.She was an exemplary TA in the real analysiscourse that I took in the same year, 1965—andover the years she also taught my nephew and twoof my children!

Jean BourgainMy early memories of Joram go back to the lateseventies and very early eighties with functionalanalysis meetings in Crete and Ohio and a work-shop at the Hebrew University. He was of coursean authority in the field, while I was a beginningresearcher at the Belgian Science Foundation. Fromour first encounters, I felt very much at ease andhappy to chat with him whenever I got a chance.He was a good person to talk to. Such things aredifficult to rationalize, but it was probably the com-bination of his encouragement, an unmistakablesharpness of mind, also in mathematics outsidehis direct expertise or interests, and, above all, theperception of a human warmth that evolved intotrue friendship and affection in later years.

As a matter of fact, even at that time, Joram asa mathematician was no stranger to me. Startingfrom the mid-seventies, my advisor, F. Delbaen atthe Free University of Brussels, had introduced meto Joram’s many contributions to Banach spacetheory and some of the problems left open by hiswork: they were the background and motivation ofmy early research. These were questions related toextreme points, a subject that has been consistentlyclose to Joram’s heart. Also the global implicationsof certain local structural properties as studied inJoram’s own thesis about extension of compactoperators. So I was most excited to meet the manin person and get some feedback. Joram expressedsome praise, although not quite as much as Ihad hoped for. He was indeed in the middle of

Jean Bourgain is IBM von Neumann Professor at the Insti-tute for Advanced Study. His email address is bourgain@math.ias.edu.

Joram and Aleksander Pełczynski, 1973.

preparing the second volume of Classical BanachSpaces, together with his long-time collaboratorLior Tzafriri, which was mostly devoted to otheraspects of the theory. This reminds me of alittle anecdote at the Heraklion meeting in 1978,when Joram made it clear that “stable Banachspaces,” a concept then freshly introduced (andconceptualizing some striking results of D. Aldous),was the most important thing of the moment andshould definitely be part of the book in writing. Heshowed great unhappiness that neither J-L. Krivinenor B. Maurey, who developed that concept, werepresent at this gathering.

Joram was focused and liked to pursue mattersin depth. In the very early eighties, he came to visitin Belgium for a few days. We went sightseeingin the medieval town of Brugge, where we spentthe afternoon viewing the paintings of the greatFlemish and Dutch masters. His taste in this matterwas different from mine, though, and I tried invein to convince him that the scenes pictured byHieronymus Bosch are fascinating. Towards theevening he told me he had a favor to ask. The favorwas simply to check P. Enflo’s recent solution ofthe invariant subspace problem. With that I guidedhim back to the train station, with only a vaguepromise.

Over the years, especially in the eighties andearly nineties during my almost yearly trips toJerusalem, Joram was invariably a marvelous host,and I truly enjoyed these visits. Our interactionand collaboration centered around questions inhigh-dimensional convexity, a topic that had beenrevolutionized by methods from the so-calledLocal Theory of Banach Spaces and in particularJoram’s joint work with T. Figiel and V. Milmanon large-dimensional Hilbertian sections, whichis one of the most seminal contributions in thefield. We worked on problems of low-dimensionalembeddings (an issue that became increasinglyimportant in theoretical computer science), fast

28 Notices of the AMS Volume 62, Number 1

symmetrization, approximations, and also on veryclassical themes such as the regularity of the Gaussmap and optimal distribution of points on spheres.Coming back to low-dimensional embeddings,perhaps one of his most influential results (jointlywith W. Johnson) is the principle of dimensionalreduction in Hilbert space, which is central inmodern data processing. Then, by the mid-ninetieswe had drifted somewhat apart as we got to workon different things. Joram had gotten back to thelove of his youth, which is infinite-dimensionaltheory, and started a fruitful collaboration withD. Preiss and J. Tiser on Fréchet differentiability ofLipschitz maps.

Through a significant part of his professionalcareer, especially in the later years, Joram had tostruggle with severe health problems. His answerto them was a motivation and determination in hiswork that is exemplary to all of us.

These are some personal reminiscences andcomments, but many of them are surely shared bymy colleagues. Of course, we all miss him.

Nassif Ghoussoub

I knew that Joram had been seriously ill for sometime, but the cryptic email announcing his passingbrought more than its share of extreme sadness.Both my professional and my personal lives havebeen deeply touched by Joram and his family. Iworked with him on several projects, and hearingthat he, as well, was gone only a few short monthsafter another friend and coauthor, William J. Davis,passed away feels like a bad dream.

But Joram was much more than a coauthor tome. I was a twenty-two-year-old “kid” when I firstmet him. It was in Columbus, Ohio. He was alreadya leader in functional analysis, my field of researchat that time. It is fair to say that he was a fearedleader, with extremely high scientific standards. Hewas tough and never minced his words, but I neverfelt intimidated by him, though I was well awarethat many other, often smarter, mathematiciansaround me were. He was quite demanding of hisstudents: he wanted them to excel, and they did.

I often wondered whether others understoodthis man the way that I did—this man who seemedso tough on the outside was so gentle, even soft,on the inside. A sabra! Confirmation came severalyears later when I got to know his children. He wasas demanding as an old-fashioned patriarch couldbe, but they knew …

Nassif Ghoussoub is professor of mathematics at the Uni-versity of British Columbia and scientific director of theBanff International Research Station. His email address isnassif@math.ubc.ca.

I was “fresh off the boat” on the Americancontinent for a postdoctoral position. Joram Linden-strauss was already a pillar of Israeli mathematics.He was the first Israeli I ever met. The encounterwas one that would mark my life. His parents hadleft Germany for Jerusalem as soon as the Naziscame to power. He was born in the holiest of places,lived there all his life, and endeavored to make itsHebrew University one of the best, and not onlyin Israel. We quarreled about politics of course,yet there was always this feeling, which may seemnaive nowadays, that all would end up well oneday. It was there and then that I first learned—and,yes, relatively late in life—that “what we have incommon is much greater and more powerful thanwhat divides us.”

All this was before I met his incredibly kindwife, Naomi, and his amazing, then-teenaged chil-dren. Watching the Lindenstrauss family togetheramounted to seeing humanity at its glorious best.That’s how I wanted my own family to be. I’vewondered lately how incredulous he, a recipientof the Israel Prize with a Fields Medalist for ason, would have been upon watching Footnote themovie. His children lived up to every expectation.He must have been so happy as he passed.

Joram’s mathematical contributions are numer-ous and varied. His defining role, with AlexanderPełczynski, in uncovering the true impact and depthof Grothendick’s “Résumé” is well documented.The Figiel-Lindenstrauss-Milman paper on “thedimension of almost spherical sections of convexbodies” is a classic. The Johnson-LindenstraussLemma about nearly isometric embeddings offinite point sets in lower-dimensional spaces isone for the ages. The depth of his latest work ongeometric nonlinear functional analysis with DavidPreiss and others defied the trends and defined anew age for the field. Joram was a mathematicaltrendsetter because he never cared whether hismathematics followed the trodden path.

Joram—colleague, mentor, friend. My life hasbeen deeply enriched by his presence.

William B. Johnson

I met Joram in 1972 when he asked me to speakat a conference. He and Olek Pełczcynski were theacknowledged leaders of the resurgence of Banachspace theory, while I was a beginning researcher,yet within fifteen minutes the great Lindenstraussasked me a math question! Although the questionwas right up my alley, I had to work most of thenight in order to have something for Joram the nextday. This was the beginning of a friendship andcollaboration that spanned five decades, and I takepride in the fact that I have more collaborations

January 2015 Notices of the AMS 29

than anyone else (fifteen according to MR) withJoram.

In the 1970s Joram’s family spent severalsummers in Columbus (I was at Ohio State then),and my family spent one year in Jerusalem. In 1981–82 both families were in College Station, whereJoram and I took our sabbaticals and Joram’s wife,Naomi, worked on her PhD in computer science.Our wives became close friends and our childrengrew up together. Once when we went for Shabbatdinner at the Lindenstrauss home, Joram, withsome help from Ayelet and Elon, built a Lego cityin their living room. Our son was always happy toreturn to “Joram’s toy store.” Later I was a coach of asoccer team on which my son and Joram’s daughterplayed, and I played basketball with Kinneret andsome of her boy friends. Kinneret, now well knownin the biophysics community, was the best playeron the court and had a successful second career asa professional basketball player. Shabbat dinnersat the Lindenstrauss home after our children weregrown were particularly enjoyable when some oftheir children were present. It was great to getto know the adult Gallia, the youngest and mostwidely read of the Lindenstrauss clan (Google herto find out why), who was often present helpingNaomi with the preparations.

Joram’s and my mathematical collaborationsranged from nonseparable Banach spaces to thegeometry of finite metric spaces. To Joram thesewere not very different. He viewed his migrationsfrom topic to topic as natural. Our early researchwas in the linear world, although even in the 1970sJoram tried to interest me in the nonlinear geometryof Banach spaces. I knew well his landmark 1964paper, in which he laid out a blueprint for whatnonlinear Banach space theory should be, butI thought I had no intuition for the topic. Thatchanged in 1981 during our sabbaticals. Marcus andPisier, as a consequence of their work on stochasticprocesses, proved a seemingly unrelated result onthe extension of Lipschitz mappings from finitesubsets of Lp, 1 < p < 2, into a Hilbert space.Their theorem suggested a general result, whereLp is replaced by a general Banach space, butbecause of the nature of their proof, they did notget a result even for the Banach space L1. Joramand I realized that we could solve the problemif we could prove a dimension reduction lemmain Hilbert space. After formulating the lemma(now called the Johnson-Lindenstrauss Lemma) weproved it in fifteen minutes. Still, we knew it wasa neat result because it not only allowed us tosolve the problem of Marcus and Pisier but alsoeliminated the “curse of dimensionality” in certainhigh-dimensional pattern recognition problems.Of course, we had no idea that this lemma wouldbecome the most quoted result of either of us,

having 1,000+ references according to GoogleScholar, more than three times the referencesfor all of our other research articles combined,and getting 134,000 hits when Googling “Johnson-Lindenstrauss lemma.” Joram’s appreciation ofthe J-L Lemma is revealed by looking at the listof his selected publications that Joram drew upin the year before his death when he knew thatthe end was near; that is, the paper containingthe lemma is not among the twenty-six articleshe selected! Actually, I was not surprised by that;Joram put a premium on difficulty and was notvery comfortable with the attention the J-L Lemmareceived.

Joram, Gideon Schechtman, and I spent a lotmore time on the geometry of finite metric spacesin the 1970s and wrote three articles on thetopic, but even after averaging these with theJ-L Lemma paper, our results per hour of workon the topic were pretty low. At the end of thedecade I thought my initial desire to stay awayfrom the nonlinear world was correct. Then inthe 1990s I explained to Joram how results in thelinear theory could combine with an argumentof Bourgain to show that certain Banach spaces(specifically `p for 1 < p < 2) are determinedby their uniform structure (a Banach space isdetermined by its uniform structure if wheneverit is uniformly homeomorphic to a Banach spaceY , it must be linearly homeomorphic to Y ). Thisexcited Joram and led to a series of papers, all jointwith Schechtman and all but one joint with DavidPreiss, on nonlinear Banach space theory in aninfinite-dimensional setting. It’s a good thing that Ilet Joram drag me into the project, as the only threeof our joint papers that appear on his selectedpublications list are from this period. (As you cansee, even after all these years I long for Joram’sapproval.) After these collaborations, David andJoram continued in the nonlinear world and diddeep research on the differentiation of Lipschitzfunctions, culminating in their book with J. Tiser.Joram was very happy with this work and waslooking forward to doing more on differentiationtheory, but, alas, that was the last mathematicalcontribution he was to make.

Ayelet LindenstraussMy father loved being a mathematician. The boywho grew up in Israel and left it for the first timeafter completing his PhD never ceased to marvelat being a member of the international fellowshipof mathematics. To us, his children, it was also awonderful thing. Some of my father’s earliest and

Ayelet Lindenstrauss is associate professor of mathemat-ics at Indiana University. Her email address is alindens@indiana.edu.

30 Notices of the AMS Volume 62, Number 1

Joram holding Ayelet; parents Ilse and Bruno.

best mathematical contacts were in Poland. Weknew of many people who were separated fromfriends by the Iron Curtain, but my father was theonly person that we knew who had made friendsacross it. By communicating through colleaguesin Western Europe, he was able to maintain thesefriendships as relations between Poland and Israelworsened and resume them when the Iron Curtainfell. My father had a Syrian friend and a Lebanese;none of my friends’ parents knew anyone fromthese countries. Mathematicians my parents talkedabout, from faraway places, would then show upin our house.

Often my father took us along on his mathe-matical trips, especially when there were beautifulplaces to be seen. When we were young and hewent on trips without us, he would hide chocolatesin various cabinets around the house for us tofind: easy ones in the kitchen cabinets for thefirst days, and hard ones in upper cabinets werarely used for the later days. When he got back,there were always lots of presents. Once a customsagent, inspecting my father’s suitcase, suspectedhim of being a toy salesman. When I was twelveand decided I wanted to embroider on evenweavefabric, my father returned from Switzerland with afull meter of the most glorious handwoven linen.I used it very sparingly and worked every last bitof it. Afterwards, when I needed kinds of threadwhich were not available in Israel, I would tell himwhat I wanted, and during his next trip he would goto a thread store and pick out colors. Apparentlyhis repeated visits caused some of the salespeopleto wonder what he did with the threads, but healways brought me very useful color ranges.

My father did not talk much about his workat home, and when I got to my second year asan undergraduate, he chose not to teach whathad become his signature second-year analysiscourse because I would be taking it. (Many ofmy classmates were quite unhappy about this.)My main mathematical interaction with him waswriting, from his outline, most of the second

volume of his (Hebrew) textbook for this analysiscourse. It is a sampler of topics in analysis, eachpursued long enough to prove a great theoremor two. My father had very high standards formathematical writing, particularly for getting tothe heart of the matter as quickly as possible andfor not writing anything in a more complicated waythan was absolutely necessary. I certainly learneda lot from the experience.

I do a very different kind of mathematics thanmy father did. He was frustrated by the distance,but maybe it is not so far fundamentally: mystarting point is also geometry. When I prepare aclass or write out a calculation, I often hear himpreferring one approach over another.

In the last thirteen years, my father thoroughlyenjoyed his new role as a grandfather. My familyand I miss him very much.

Elon LindenstraussMy father influenced me in many ways—some thatI am aware of, some that I am not. Mathematics hasbeen tangibly present in my parents’ house sinceI can remember. Many dinnertime conversationswould be about academics; from time to time,particularly when a collaborator of my fatherwould come for an extended visit, a mathematicianwould join the family for an informal dinner.Sometimes there would be more formal dinnerswhere we kids stayed in our rooms and helped (or atleast tried not to hinder) my mother’s preparations.One of my mother’s favorite stories is that whenErdos came to a party at my parents’ house whileon sabbatical in the US, he immediately wanted tosee us, the Epsilons, who were upstairs in a part ofthe house that we (which probably mostly meansmy mother) had not had time to tidy up.

While academic life was frequently mentionedin our house, my father did not talk much abouthis mathematical work. He never pressured us inany way to become mathematicians, though bothmy big sister Ayelet and later I decided to go in thisdirection. Indeed he seemed initially ambivalentabout mathematics as a career choice, even thoughhe clearly enjoyed being a mathematician. He wascertainly extremely pleased to follow our progress.

As a child I played board games with my father.Risk and Othello (aka Reversi) were among ourfavorite board games, and as in all endeavors,he was very systematic and thorough, developingstrategies that he shared with me. When I got olderand started getting interested in mathematics, hewould suggest books from his large and carefullyselected mathematical library. Only once haveI been directly taught by him—in a course for

Elon Lindenstrauss is professor of mathematics at HebrewUniversity. His email address is elon@ math.huji.ac.il.

January 2015 Notices of the AMS 31

mathematically inclined youth, where he wouldpresent problems and we would try to solve them.In retrospect I am very happy for this experience.

My father was very honest, had high standardsboth for himself and for others, and always saidexactly what he thought, regardless of whetherwhat he thought was pleasant or unpleasant tohear. From the time I was a graduate student tothe last talks of mine which he attended when Iwas already a well-established researcher, I wasalways especially nervous and tried to be very wellprepared when giving a talk when he was in theaudience, as I was sure I would be told afterwardsexactly what he thought of the talk.

Until very close to his death, even as his healthwas deteriorating, I have relied on his advice, bothon mathematical and nonmathematical matters.He was a source of pride and strength to our familyand will be dearly missed.

Vitali Milman

It was the ICM-1966 in Moscow. A lot of mathe-maticians arrived from the West, but my highestexpectation was to meet Dvoretzky and Linden-strauss. I knew well one of the first papers byJoram about duality for the moduli of convexityand smoothness and also read all of his workthat I could find in our (poor) libraries. However,Dvoretzky indeed arrived, but Lindenstrauss didnot. Dvoretzky told me that “they” (Russian au-thorities) wrote to Joram that there is no room inhotels (?!) and they cannot let him in. So, the firsttime I met Joram was in Israel in 1973 after myemigration. It was a very difficult time, after theYom Kippur war. My family stayed in a dormitoryfor new emigrants in Tel Aviv. Once someoneknocked on our door. I opened and saw a young,extremely nice-looking person who looked at meand said, “Joram Lindenstrauss.” I remember thismoment well after forty years. I lost my voice,and I hardly remember the continuation of ourfirst meeting. Despite hundreds of days we spenttogether later, and despite the passing of fortyyears, that first image of Joram stays in my mind,comes to my mind when he is mentioned, and isnot shadowed by later changes.

Vitali Milman is professor emeritus of mathematics atthe University of Tel Aviv. His email address is milman@post.tau.ac.il.

Joram and Elon at IMU, 2006.

Our serious scientific cooperation started twoyears later (I needed this time to learn Hebrewand English, at least to understand a little bit ofboth) and resulted in joint papers with Figiel (Figiel-Lindenstrauss-Milman) in 1976 (Bulletin AMS ) and1977 (Acta Math.). I heard opinions that thesewere the most significant results in geometricfunctional analysis in the 1970s. I learned a lotfrom working on this paper with Joram, learningfrom his broad knowledge and his taste. I feltthat I became a different mathematician at theend of this period. Unfortunately, this cooperationstopped and returned only ten years later. Weactually prepared some directions and ideas forworking together, and I even wrote a few pagesof notes. But one young mathematician heardthe discussion on these results, quickly wrote apaper on them, and submitted it. Joram was veryangry, for the “cornerstone” for a new direction wewanted to build was taken out from under us, andour cooperation was stopped for a long decade.

Our second period of research cooperation, fromthe mid-1980s, was joint with Jean Bourgain. It also,I think, was very successful. That time I turnedto the direction of convexity, but “asymptotic”convexity, not the classical one, and “pushed”Joram to discuss this subject during our summerstays at IHES. I hope he liked the outcome as muchas I liked it.

Our joint activities and cooperation were notreduced to joint research. From the start of the1980s we organized a seminar (mostly in Tel Aviv)on geometric aspects of functional analysis, whichsoon became very famous and world known underthe nickname GAFA seminar. For many years it metregularly, generally twice monthly on Fridays, andattracted a lot of people from all over Israel (andmany foreign guests). Six books of proceedingsof this GAFA seminar were published during thattime, mostly by Springer, jointly edited by the twoof us. Later, the health of Joram did not allow him

32 Notices of the AMS Volume 62, Number 1

to come regularly, and the seminar changed itsappearance.

In his work and his activity, Joram alwaysemphasized nontriviality and difficulties, but alsoquickly caught new ideas and had good taste. Hedid not allow “easy” works to come through hishands. This harsh approach of his kept the highlevel of research in geometric functional analysisand also had a great influence on the Israel Journalof Mathematics during the period he was a leadingeditor.

The loss of Joram is a great loss to all of us—hiscolleagues, friends, and mathematics in whole.

Assaf NaorBeing the last doctoral student that Joram advisedbefore retiring, I have known him for a shorterperiod than the other contributors to this memorialarticle. For this reason I will not describe a personalstory about Joram, but rather mention aspects ofhis impact on mathematics and mathematicians.

Joram’s influence was multifaceted. I and manyothers who interacted with Joram associate hisname with uncompromising professional stan-dards, be it integrity, good taste in choosingresearch projects, or reserving praise only for re-sults that contain truly outstanding and importantnew ideas. This approach inevitably made Joraman exceptionally sharp critic, perhaps somewhatintimidating at times, and always an inspiring rolemodel.

Joram’s mathematical contributions were ex-emplified by deep and original insights combinedwith remarkable feats of technical strength. Thisresulted in his solution of some of the oldestand most important questions on the geometryof Banach spaces. In addition, Joram had transfor-mative impacts on mathematics by putting forthnew research paradigms that shifted the focusof subsequent work, and after decades of effortsby Joram and others, his deep insights have ledto rich new theories of central importance. Anexample of Joram’s forward-looking introductionof a powerful research agenda is his work on thenonlinear geometry of Banach spaces, motivatedby rigidity phenomena (some of which were hisown discovery) that indicated that there shouldbe a “dictionary” that translates insights from thegeometry of Banach spaces to the setting of generalmetric spaces. Almost fifty years after his initialcontributions along these lines, one can safely saythat this approach has led to many unexpectedresults in metric geometry, spreading the influenceof ideas that originate in Banach space theory toareas such as computer science and group theory.

Assaf Naor is professor of mathematics at Princeton Univer-sity. His email address is naor@math.princeton.edu.

These cross-cutting links between mathematicaldisciplines were far from obvious when Joraminitially formulated his questions, requiring (inhindsight) coping with new phenomena and theintroduction of new tools that go far beyond whatwas previously understood in the linear theory.As an example, one can point out Joram’s re-markable intuition, formulated jointly with W. B.Johnson, that there should be a nonlinear analogof Maurey’s extension theorem (a phenomenonthat was eventually verified due to ideas of K. Ball).This was put forth in his work on extension ofLipschitz functions, a paper that included, as atool, a dimensionality reduction lemma which hashad an extremely important and central impact onvarious aspects of computer science (exemplifyingJoram being a sharp critic, he believed that thislemma, despite being his most cited work, was toosimple to be considered an actual result).

Joram’s death is a huge professional and per-sonal loss to many people. It is certain that hisinsights and profound impact on mathematicswill perpetually endure and even increase overtime as more progress is made on his long-termresearch programs and the ideas and methods thathe introduced are used in new contexts.

Gilles PisierReminiscing about Joram, the first thing that comesto mind is how incredibly nice and supportivehe was to me early on. So much so that theinitial awe that I had of him quickly evaporated,even though we never exchanged too many words.In fact, although this cannot be entirely true, Iremember having only “serious” conversationswith him, meaning all revolving around math, insharp contrast with the discussions I had with hisfriend and colleague Lior (Tzafriri), which coveredthe whole spectrum and could be at times veryfunny or quite intimate. Joram always remained(mostly in my imagination) a rather tough fatherfigure always in demand for deeper and hardertheorems.

I met Joram for the first time in Oberwolfachin October 1973. He and A. Pełczynski emergedthere as the two main leaders of the new field tobe labeled “geometry of Banach spaces.” I was onlytwenty-two and he still thirty-six for a few moreweeks. This was also the first time I met most of myfuture friends and colleagues in that field, includingTadek Figiel, with whom conversations led to atheorem that to our surprise (because it lookedto us as a mere combination of known results)Joram pushed us to publish together. This was

Gilles Pisier is A. G. and M. E. Owen Chair and DistinguishedProfessor at Texas A&M University. His email address ispisier@math.tamu.edu.

January 2015 Notices of the AMS 33

an isomorphic characterization of Hilbert spacesas those Banach spaces E admitting an equivalentnorm with (essentially) best possible modulus ofconvexity δE and one (possibly different) with bestpossible modulus of smoothness ρE . Joram hadlong been interested in questions related to thesenotions and had proved a famous duality formulabetween δE and ρE∗ .

We met again in Durham in summer 1974when I presented a more substantial theoremon renorming of uniformly convex spaces usingmartingales. I don’t remember any comment fromhim, but he and Lior immediately invited me to visitthem in Jerusalem, which I did for two months inDecember 1974. While in Durham, Joram heard thatI was preparing a paper with Per Enflo on what wecalled the 3-space problem: If a given Banach spaceX has a closed subspace Y ⊂ X such that both Yand X/Y are isomorphic to Hilbert space, is it alsotrue for X? This was a famous question attributedto Palais (but this was never confirmed). Enflo andI could not solve it but showed that in terms oftype, cotype, uniform convexity and smoothness,the space X was very close to being Hilbertian.For instance, in the n-dimensional version of theproblem we concluded that X was Cn-isomorphicto `n2 with Cn = O((logn)α) for some α > 0. Joramtold me he had an approach to that problem and,to my terror, insisted that I give him a privatebriefing to describe our results, at the end ofwhich he made no comment, but I remember beingpuzzled that he seemed happy. Perhaps it madehim feel he was on the right track. Later on thatsame year but in Jerusalem, Joram showed mehis counterexample to the Palais problem, andto my surprise (and actually against my will!) hecontacted Enflo to stop the publication of ourpaper (on which printer composition had started)in order to make sure that he could join us as acoauthor. In Joram’s ingenious counterexample,the distance to Euclidean n-space was larger thanc√

logn. Thus it showed that the positive resultswere essentially best possible, and so the finaljoint paper gave a quite complete solution of thePalais problem. This took me by surprise and tothis day I remember feeling embarrassed to havebecome, at his insistence, a coauthor of such amajor breakthrough. Kalton and Peck later gave adifferent example (usually denoted by Z2), whichis very closely related to complex interpolationtheory for the pair (`1, `∞). Kalton also gave anexample where the above

√logn is replaced by

logn, which is sharp.From the 1973 meeting I remember with emo-

tion Bob James’s lectures on his nonoctahedralnonreflexive outstanding example, solving a long-standing problem. In a famous Annals paper fromthe 1960’s he had proved that any nonreflexive

Joram receives the Israel Prize, 1981.

space must be square. A space is called octahe-dral (resp. square) if it contains for any ε > 0 a(1 + ε)-isomorphic copy of `3

1 (resp. `21). James

went on for hours in his own strange style of“hands-on” mathematics, seemingly allergic to themore commonly formalized statements otherswere used to, and everybody seemed lost save forJoram. James had made premature claims in thepast decade, so there was some skepticism in theaudience. To us junior auditors he looked like hehad hit too many a wall and all he kept doing wasexplaining how to cleverly add “bumps,” but thistime he was right! Joram listened patiently till theend, and the next year he produced with Jamesan improved and much more digestible version ofJames’s landmark example. Moreover, with Davisand Johnson he developed a penetrating analysisof the degrees of reflexivity of a Banach space.A sample result is that while James’s exampleshows that nonreflexivity of a space X fails toimply that X is octahedral, the nonreflexivity ofR(X) = X∗∗/X does. This somewhat explains whythe James examples were found among spaces ofcodimension 1 in their bidual (i.e. dim(R(X)) = 1),just like the space J for which James had be-come famous in the 1950’s. More precisely, lettingR2(X) = R(R(X)) and Rk(X) = R(Rk−1(X)), theyshowed that if Rk(X) 6= 0 (i.e. Rk−1(X) is notreflexive), then for any ε > 0, the space X containsa (1+ ε)-isomorphic copy of `k+1

1 .This reminds me of a very dear souvenir of the

Lindenstrauss family, a sort of personal treasure.In the type/cotype language, the above examplesshowed that type p > 1 did not imply super-reflexivity (or reflexivity). Thus it was natural towonder what happened in the extreme case p = 2.Joram knew that I was obstinately trying to provethat type 2 implied reflexivity while in Jerusalemback in early 1975, and one day he called to saythat he thought he had a proof using the iteratedlogarithm law (this later collapsed), but since his

34 Notices of the AMS Volume 62, Number 1

son, Elon, was sick and his wife, Naomi, had to goteach, he had to stay at home, so would I agree tocome to his place to talk while babysitting with him.Of course I was delighted, so he picked me up andwe worked for a couple of hours at his home untilNaomi’s return. How could I have imagined thenthat the charming little toddler who was jumpingaround under our table was a future 2010 FieldsMedalist!

To complete the James saga, shortly afterOberwolfach, Joram and Bill Davis showed theexistence among the James zoo of nonreflexivespaces of type p for any p < 2, and finally Jameshimself showed (while we were all back in Jerusalemfor a special year in 1976/77) that even type 2 doesnot imply reflexivity. Eventually, a few years later,Xu and I managed to exhibit (by a different method,using real interpolation), for all 1 ≤ p ≤ 2 ≤ q <∞,nonreflexive examples with type p and cotypeq, except for the obvious exception p = q = 2characterizing Hilbert space.

David PreissI feel as if I have known Joram all my life, evenif for the first forty years I lived in a countrywhose relations with Israel were not on a level thatwould allow us to work together, and even ourone brief meeting at a conference was not to bementioned too loudly. My mathematics, however,has always been deeply influenced by Joram’swork, as documented by a referee calling me “amathematician of Lindenstrauss’s school” longbefore the political situation allowed us to meetand work together. Towards the end of the eightiesNassif Ghoussoub considered inviting me for avisit and asked Joram about me. Joram respondedthat we had never met (and then he said somethingpositive about my mathematics). When I later toldhim that we actually had met, he said, “So I waswrong” (specifying quickly “but only in one point”).I remember this because of my admiration for his“so I was wrong”: it was very rare, because he wasusually right; but when he wasn’t, he would notwaste time arguing. Just these four words, and wewould start discussing new ideas.

I recall a number of nonmathematical events,such as playing basketball on a team opposing him(he was very good), going to concerts, his daughterGallia’s pictures of Jerusalem, walks in Jerusalem,exhibitions, and of course his welcoming familyand the huge amount of help we were givenby his wife, Naomi. Surprisingly, I cannot recallwhen we started talking serious mathematics. Mostprobably, we began by discussing the Lipschitz

David Preiss, FRS, is professor of mathematics at theUniversity of Warwick. His email address is d.preiss@warwick.ac.uk.

isomorphism problem and its possible solutionusing derivatives. This still-open problem askswhether, say, reflexive, separable Banach spaces Eand F are linearly isomorphic provided they areLipschitz isomorphic. A natural candidate forthe linear isomorphism is the derivative of theLipschitz isomorphism at a suitable point. Inspecial situations the use of Gâteaux derivatives,which are known to exist but need not be surjective,combined with results from the geometry of Banachspaces gives a positive answer. Fréchet derivativeshave the property that the derivative of a Lipschitzisomorphism is a linear isomorphism, but evenLipschitz self-maps of Hilbert spaces can fail to beFréchet differentiable at any point. Nevertheless,the Gâteaux derivative of a Lipschitz isomorphismof E to F at a point x is a linear isomorphismprovided that its compositions with elements of thedual of F are Fréchet differentiable at x. Whethersuch a point x exists (when E, F are reflexive, say)is open. This problem is naturally divided into twoparts: firstly, whether any countable collection ofreal-valued Lipschitz functions on a reflexive spacehas a common point of Fréchet differentiability,and secondly, whether such a point can be a pointof Gâteaux differentiability of a vector-valuedfunction.

Much of what we did with Joram was related tothe above problems. Our first paper, written mostlyduring my first longer-term visit to Jerusalem, wasmotivated by the observation that a weakeningof Fréchet differentiability, called almost Fréchetdifferentiability, suffices to answer the Lipschitzisomorphism problem. We constructed such pointsfor any finite number of real-valued Lipschitzfunctions on superreflexive spaces. We spent sometime discussing whether the method, a variant ofdensity points, could lead to stronger results, buteventually agreed that there are serious obstaclesto it. In the end, in a paper with Johnson andSchechtman, we gave a much simpler proof of thisresult from which the main obstacle to extendingit to countably many functions is clearly seen.

When we began our investigation of differentia-bility problems, it was known only that real-valuedLipschitz functions on spaces with separable dualhave points of Fréchet differentiability. It wasnot even clear whether there is a single infinite-dimensional Banach space in which any two suchfunctions have a common point of Fréchet differen-tiability. The key problem in trying to find such apoint seems to be that the (weaker) Gâteaux differ-entiability requires a measure-theoretic concept ofsmallness, while the Fréchet requirement is closerto the use of Baire category. Mixing measure andcategory smallness are fraught with the dangerof proving excellent results for elements of theempty set. Nevertheless, we noticed that while

January 2015 Notices of the AMS 35

measure theoretical smallness is required in thespace, smallness in the sense of category may beneeded only in a suitable space of measures. Thuswe defined a new class of negligible sets, whichwe called Γ -null sets, as those sets that are nullfor typical (in the sense of category) measures ina naturally chosen space of measures. In someBanach spaces, including c0 and Tsirelson’s space,we showed that real-valued Lipschitz functions areFréchet differentiable Γ -almost everywhere, and forthese spaces we therefore know that every count-able collection of such functions has a commonpoint of Fréchet differentiability. However, we alsoproved that with the space of measures that wehave used or with similar spaces this program failsin Hilbert spaces.

An important point was our recognition thatFréchet differentiability problems are sharply di-vided into two categories depending on whetherone requires validity of mean value estimates or not.The validity of mean value estimates means that theclosed convex hull of the set of Fréchet derivativesand of Gâteaux derivatives coincide. In an “onand off” (as Joram called it) investigation lastingabout ten years, joined also by Jaroslav Tišer fromPrague, we found reasonably satisfactory answersin the first case. Perhaps the easiest of our resultsto state is that R2-valued functions on Hilbertspaces have so many points of Fréchet differen-tiability that the mean value estimates hold, butR3-valued Lipschitz functions may have so smalla number of them that the mean value estimatefails, and analogous results hold for n functionson `n. Our rather involved proofs appeared inthe research monograph Fréchet Differentiabilityof Lipschitz Functions and Porous Sets in BanachSpaces, published in February 2012. We did notconsider this as the end of our joint work, butcorresponded about questions we should study inorder to understand differentiability without meanvalue estimates; better understanding of Gâteauxdifferentiability featured most prominently. Un-fortunately, Joram’s health did not allow us tocontinue these discussions.

Gideon SchechtmanI first met Joram when I was an undergraduatestudent at The Hebrew University in Jerusalem. Itook Introduction to Functional Analysis from him,but strangely enough I decided to try to recruit himas my PhD advisor only after I had a magnificentcourse in topology with him, an area which wasto a large extent new to Joram as well. Until thenI was seriously thinking of completely different

Gideon Schechtman is professor of mathematics at theWeizmann Institute of Science. His email address isgideon@weizmann.ac.il.

directions of research (first game theory, then settheory). When I approached Joram and asked himto be my PhD advisor, he was already advising atleast four PhD students, and I got the (probablycompletely unjustified) impression that he wastrying to discourage me. He gave me three papersto read. The first was Dvoretzky’s spherical sectionof convex bodies paper, an extremely importantpaper that only a handful of people managed toread. I’m not one of them even to date. The secondwas a paper on the geometry of Lp spaces thatwas very badly written and in a language I wasnot fluent in. It contained a very nice result, and Imade a great effort to understand it but failed. Itlater turned out that both the proof and the resultwere wrong. I don’t remember the third paper.

I don’t know if it was the intention of Joram,but actually these two papers, in spite of the factthat they caused me such frustration (and maybebecause of that), had a major influence on my laterresearch. Most of my PhD thesis revolves aroundthe geometry of Lp spaces, and some of my laterresearch is very much connected with Dvoretzky’stheorem. I remember vividly the few encouragingwords that I heard from Joram after I provedmy first result. I guess this was the first time Iheard some compliments from him, and I mustadmit that even many years later I was strugglingwith mathematics mostly to squeeze a few goodcomments from him (I’m exaggerating a bit, butthis is not far from the truth). During my PhDyears Joram and I tried to cooperate twice. One ofthese periods came after I thought I had solvedthe distortion problem (that was solved almosttwenty years later by Odell and Schlumprecht), butnothing written came out of it. Yet, I still learned alot from these experiences. Also, more importantly,it helped me overcome a fear of him, and little bylittle we became colleagues rather than a masterand a student. Our first fruitful cooperation cameonly a few years later.

So what was Joram’s main influence on me? Ican’t say it is the direction of research he set meto; I would probably follow him in any directionhe would be in. Also, I cannot honestly say thathis mathematical power and insight (which wereundoubtedly great) had a unique influence on me.What really set him apart from the point of view ofhis influence on me were the standards he set andlived by. He had very high standards as to whatconstitutes good mathematics and for scientificand personal integrity. These were also very easyto absorb from him: he always said what was onhis mind whether asked for it or not. Until late inmy career a major component in my decision ofwhether or how to write a paper on a result I gotwas trying to estimate what Joram’s opinion of itwould be and trying to avoid a sarcastic remark

36 Notices of the AMS Volume 62, Number 1

Joram and wife, Naomi, in Switzerland, 1994.

from him. On the other hand, any expressionof appreciation from him had double the value,because he never said something different fromwhat he really thought.

Andrzej SzankowskiJoram was for me a very important person; I thinkhe influenced my life more than anyone outside myfamily. Although he was not my official advisor, hehas been for me the main guidance in mathematics.

I first met Joram in Aarhus in spring 1970. UntilI settled in Israel in 1980, every year I visited himin Jerusalem. During my first visit in December1970, I witnessed a spectacular display of Joram’sabilities. He worked then, with Lior Tzafriri, on thecomplemented subspaces problem. A completesolution seemed out of reach, and the main effortwas to settle it in special cases. And then onemorning Joram came beaming to the institute andsaid “I worked on it the whole night and here isthe solution." So it was—a very clever, clear-cut,complete solution to a major problem. This wasdone with a “feedback technique” which Jorammastered: in order to prove that a quantity isbounded, it suffices to obtain an inequality inwhich it is bounded by a decreasing function ofitself.

The 1970s were the golden period of Joram’sseminar in Jerusalem. Almost every week he orone of his students came up with a new result. In1976–77 Joram organized a Banach spaces year atthe Institute of Advanced Studies of the HebrewUniversity. This was a great year for most of theparticipants, for me in particular: day by day Joramand I discussed the problem of the approximation

Andrzej Szankowski is professor emeritus of mathematicsat the Hebrew University. His email address is tomek@math.huji.ac.il.

property for B(H). This problem was somewhatabove my limits, and I don’t think I could havecoped with it without Joram’s encouragement andadvice.

In the 1980s I collaborated with Joram onseveral problems. We have a nice paper in nonlinearanalysis which came up in a typical (for Joram)way: he got me interested in a paper from whichI observed a generalization of the Mazur-Ulamtheorem; after seeing it, Joram overnight figuredout how to turn it into an “if and only if” theorem,a complete result. Another paper from this periodperhaps would not be remarkable, except that itcontains probably the first contribution of Elonto mathematics: Elon, then a high school student,wrote for us a computer program which computedthe maximum of a monstrous function that cameup from our computations.

Joram was a great lecturer. He had a veryspecial style: writing rather little on the black-board, he made an impression of improvising, but,miraculously, everything was very well organized.I think it came quite naturally to him, withoutmaking notes or much preparation. He wrote twobeautiful textbooks in Hebrew: Advanced Calculusfor second-year [undergrad students] and Introto Functional Analysis (together with A. Pazy andB. Weiss), for MSc. students.

From the moment I first met Joram, I came to likehim as a person. He had sharp opinions and wasn’treluctant to put them forward. This made him apoor politician but a valuable teacher and friend.Joram set high standards for what was publishable,both for himself and for his students. He was highlysuspicious about general theories which didn’tseem to have interesting models. Joram was a verymodest man, handling in a relaxed way numeroushonors which were bestowed on him. He was verycareful with superlatives, rather consciously usinghis own scale of appreciation, in which “beautiful”or “deep” were used very seldom, and “profound”was apparently reserved for Dvoretzky’s theorem.

Joram’s world was centered about two subjects:mathematics and his family. He had many friendsaround the world, but as he used to say, thesewere either mathematicians or family. He was asocial man and enjoyed entertaining people. I havefond memories of many evenings I spent at Naomiand Joram’s home. He took special care of youngpeople, not only his own students. Inviting themfrequently, together with influential people, hehelped them to establish contacts in a natural way.He obviously took care of his students well beyondthe expected.

Joram’s last years were marred by his dete-riorating health, but he must have felt a senseof fulfillment, both as a mathematician and as afamily man.

January 2015 Notices of the AMS 37