Every Waking Moment Ky Fan (1914–2010) Bor-Luh Lin Ky Fan passed away on competition provided one student with a chance to March 22, 2010, at the age pursue a degree in mathematics in Europe. Work- of ninety-five in Santa Bar- ing with Maurice Fréchet, he received his D.Sci. bara,California. He was born from the University of Paris in just two years, with in Hangzhou, China, on Sep- a thesis with the title Sur quelques notions fonda- tember 19, 1914. He enrolled mentales de l’analyse générale. He was a French in National Peking Univer- National Science Fellow at Centre National de la sity in 1932. Despite an Recherche Scientifique in 1941–1942 and a mem- interest in engineering, he ber of the Institut Henri Poincaré in 1942–1945. pursued studies in mathe- By 1945 he had already published twenty-five pa- matics due in part to the pers on abstract analysis and topology, including influence of his uncle, Zuxun the monograph Introduction à la topologie combi- Feng, who was chair of the natorire, I. Initiation (Vubert, Paris), written with Department of Mathematics M. Fréchet. at Peking University. As a Fan was at the Institute for Advanced Study Photographs courtesy of the author. junior in college, Fan was in- at Princeton in 1945–1947. As an assistant of Ky Fan spired by a visit of E. Sperner John von Neumann, and inspired by H. Weyl, and translated into Chinese he developed an interest in operator theory, the book by O. Schreier and E. Sperner, Ein- matrix theory, minimax theory, and game the- führung in die Analytische Geometrie und Algebra. ory. Later he extended his interests to systems This translation, published in 1935 with the ti- of inequalities and fixed point theory. At that tle Analytical Geometry and Algebra I, II, became a time many in these fields were studying finite standard textbook in China. In fact, in 1953, almost games and optimizations in finite-dimensional twenty years after its initial publication, I used the spaces. Fan, however, pioneered the fields of infi- same text as an undergraduate at National Taiwan nite games and inequalities in infinite-dimensional University for a course on advanced geometry. linear spaces. He made fundamental and ground- This book so inspired me that I sought to join Fan breaking contributions and continued to be a at the University of Notre Dame to pursue a Ph.D. leader in these areas throughout his career. He in mathematics. By the time Fan graduated from also made major contributions to the geometry Peking University in 1936, he had also translated of Banach spaces, convex analysis, combinato- a book of Landau on number theory and ideal rial topology, topological groups, and analytical theory and had coauthored with a colleague a function theory. book on number theory. He was an instructor at Fan taught at the University of Notre Dame Peking University in 1936–1939. In 1939 he was from 1947 to 1960. For many summers dur- selected by the China-France Education Founda- ing this period, he was at the National Bureau tion to receive a Boxer Scholarship. This national of Standards, Oak Ridge National Laboratory, and Argonne National Laboratory. He taught at Bor-Luh Lin is professor of mathematics at the University Wayne State University (1960–1961), Northwest- of Iowa. His email address is [email protected]. ern University (1961–1965), and the University of 1444 Notices of the AMS Volume 57 , Number 11 California at Santa Barbara starting in 1965. Upon from the rest of the world, as well as grants to his retirement from UC Santa Barbara in 1985, an North American departments to bring in visitors international symposium was held to celebrate his from China. The program also supports occa- many contributions to mathematics. The sympo- sional conferences in China and improvement of sium resulted in the publication of a monograph library holdings in Chinese institutions. The Fan with the title Nonlinear and Convex Analysis: Pro- Endowment has also provided grants to assist ceedings In Honor of Ky Fan (Lecture Notes in Pure programs in the United States that nurture math- and Applied Mathematics, Dekker, 107(1987)). He ematically talented high school students and has was chair of the Department of Mathematics at UC supported the Ky and Yu-Fen Fan Scholarships Santa Barbara in 1968–1969. He held visiting posi- within the AMS Epsilon Scholarships Program for tions at the University of Texas-Austin, Universität high school students. As noted by AMS past presi- Hamburg, the Université Paris IX, and Università dent Felix Browder, “The impact of Ky and Yu-Fen’s degli Studi di Perugia. He was elected a member of generosity will be felt for years to come.” Academia Sinica in 1964 and served as the director Fan published about 130 pa- of the Institute of Mathematics at the Academia pers, many of which made Sinica from 1978–1984. Two issues of the Bulletin fundamental contributions to of the Institute of Mathematics, Academia Sinica— several fields in pure and ap- V. 2, No. 2, 1974, and V. 3, No. 1, 1975—were plied mathematics. From the dedicated to him for his sixtieth birthday. In 1989 famed Ky Fan inequalities he was a Distinguished Visiting Scholar at the (1951) to the Fan condition Chinese University in Hong Kong and received an Honorary Professorship from Peking University. (1956), to the Ky Fan minimax After the visit, he donated his whole collection inequality (1972) his papers of mathematics books and treatises to Peking shaped the fields of linear and University. In 1990 he was awarded the degree nonlinear functional analysis, Docteur Honoris Causa from the Université de linear algebra, convex anal- Paris-Dauphine and was the featured speaker at ysis, and optimization. Fan the Conference on Directions in Matrix Theory had a knack for identifying (Fourth Auburn Linear Algebra Conference). His the central problems in a lecture at the Auburn conference, his publication field of study and present- lists up to 1992, a brief bibliography, and a list of ing them in the most general his Ph.D. students appeared in Linear Algebra and concise statements, with new its Applications 162–164:1-2(1992), 1–22. In 1993 approaches and elegant proofs. Ky Fan, China, the T.I.Tec/K.E.S. Conference on Nonlinear and He brought seeming unrelated 1939. Convex Analysis in Tokyo was dedicated to Fan areas together to create new in recognition of his fundamental contributions mathematics. As a result, his papers were often to the field. In 2011 the seventh International cited and pointed to new research directions. His Conference on Nonlinear and Convex Analysis in contributions concerning fixed points and min- Hirosaki will be dedicated to his memory. In 1994 imax inequalities have had a major impact in an entire issue of Topological Methods in Nonlinear the development of nonlinear functional analy- Analysis was dedicated to him for his eightieth sis. He made significant contributions to locations birthday. He served on many editorial boards. He and identified maximum eigenvalues of matrices. was a distinguished editor of Linear Algebra and His works found wide applications to mathemat- its Applications, and he was one of the founding ical economics, differential equations, potential editors of the Journal of Mathematical Analysis theory, and numerical analysis. His research and Applications and the Journal of Nonlinear and demonstrated the beauty of combining pure and Convex Analysis. The latter will publish a special applied mathematics. There are a large number issue in memory of Fan. of theorems, lemmas, inequalities, equalities, con- In 1999 Ky Fan and his wife, Yu-Fen Fan, made ditions, norms, etc., that bear the name of Ky a gift of approximately US$1 million to the Amer- Fan. The bibliography lists some major reference ican Mathematical Society. The fund was used to books that contain many of his contributions. We establish the Ky and Yu-Fen Fan Endowment to now briefly discuss the Ky Fan inequality, the Fan support and to foster collaborations between Chi- condition, and the Ky Fan minimax inequality. nese mathematicians and mathematicians in other Ky Fan Minimax Inequality (K. Fan, A minimax parts of the world, especially North America, and to support mathematically talented high school inequality and applications, Inequalities III, Proc. of students in the United States. The AMS uses the the Third Symposium on Inequalities, Acad. Press fund from the Fan Endowment to fund the China (1972), 103–113). Exchange Program, which provides grants to Chi- Let X be a nonempty compact convex set nese mathematics departments to bring visitors in a Hausdorff topological vector space E. December 2010 Notices of the AMS 1445 Suppose f is a real-valued function defined Fan Condition (K. Fan, On systems of linear in- on X X such that: equalities, linear inequalities and related systems, (a)× For each fixed x in X, f (x, y) is a Annals of Math. Studies, Princeton Univ. Press, vol. lower semicontinuous function of y 38 (1956), 99–156). on X, Let X be a real linear space. For any linear (b) For each fixed y in X, f (x, y) is a functionals fi, i 1, 2,...,n, and real numbers = quasi-concave function of x on X. ci, i 1, 2,...,n, there exists x X such that = ∈ Then the minimax inequality fi (x) ci , i 1, 2,...,n, if and only if for any non- ≥ = negative numbers ai , i 1, 2,...,n, the relation min max f (x, y) sup f (x, x) n n = y X x X ≤ x X i 1 ai fi 0 implies i 1 aici 0.