Stein was born in Antwerp Belgium, to Elkan Stein and Chana Goldman, Ashkenazi Jews from Belgium.^[1] After the German invasion in 1940, the Stein family fled to the United States, first arriving in New York City.^[1] He graduated from Stuyvesant High School in 1949,^[1] where he was classmates with future Fields Medalist Paul Cohen,^[2] before moving on to the University of Chicago for college. In 1955, Stein earned a Ph.D. from the University of Chicago under the direction of Antoni Zygmund. He began teaching at MIT in 1955, moved to the University of Chicago in 1958 as an assistant professor, and in 1963 became a full professor at Princeton.

Stein worked primarily in the field of harmonic analysis, and made contributions in both extending and clarifying Calderón–Zygmund theory. These include Stein interpolation (a variable-parameter version of complex interpolation), the Stein maximal principle (showing that under many circumstances, almost everywhere convergence is equivalent to the boundedness of a maximal function), Stein complementary series representations, Nikishin–Pisier–Stein factorization in operator theory, the Tomas–Stein restriction theorem in Fourier analysis, the Kunze–Stein phenomenon in convolution on semisimple groups, the Cotlar–Stein lemma concerning the sum of almost orthogonal operators, and the Fefferman–Stein theory of the Hardy space ${\displaystyle H^{1}}$  and the space ${\displaystyle BMO}$  of functions of bounded mean oscillation.

He wrote numerous books on harmonic analysis (see e.g. [1,3,5]), which are often cited as the standard references on the subject. His Princeton Lectures in Analysis series [6,7,8,9] were penned for his sequence of undergraduate courses on analysis at Princeton. Stein was also noted as having trained a high number of graduate students. According to the Mathematics Genealogy Project), Stein had at least 52 graduate students—including the Fields medalists Charles Fefferman and Terence Tao—some of whom went on to shape modern Fourier analysis.

His honors included the Steele Prize (1984 and 2002), the Schock Prize in Mathematics (1993), the Wolf Prize in Mathematics (1999), and the National Medal of Science (2001). In addition, he had fellowships to National Science Foundation, Sloan Foundation, Guggenheim Foundation, and National Academy of Sciences. Stein was elected as a member of the American Academy of Arts and Sciences in 1982.^[3] In 2005, Stein was awarded the Stefan Bergman prize in recognition of his contributions in real, complex, and harmonic analysis. In 2012 he became a fellow of the American Mathematical Society.^[4]

In 1959, he married Elly Intrator,^[1] a former Jewish refugee during World War II.^[5] They had two children, Karen Stein and Jeremy C. Stein,^[1] and grandchildren named Alison, Jason, and Carolyn. His son Jeremy is a professor of financial economics at Harvard, former adviser to Tim Geithner and Lawrence Summers, and served on the Federal Reserve Board of Governors from 2012 to 2014. Elias Stein died of complications of lymphoma in 2018, aged 87.^[6]

• Stein, Elias (1970). Singular Integrals and Differentiability Properties of Functions. Princeton University Press. ISBN 0-691-08079-8.
• Stein, Elias (1970). Topics in Harmonic Analysis Related to the Littlewood-Paley Theory. Princeton University Press. ISBN 0-691-08067-4.
• Stein, Elias; Weiss, Guido (1971). Introduction to Fourier Analysis on Euclidean Spaces. Princeton University Press. ISBN 0-691-08078-X.
• Stein, Elias (1971). Analytic Continuation of Group Representations. Princeton University Press. ISBN 0-300-01428-7.
• Nagel, Alexander (1979). Lectures on Pseudo-differential Operators: Regularity Theorems and Applications to Non-elliptic Problems. Princeton University Press. ISBN 978-0-691-08247-9.^[7]
• Stein, Elias (1993). Harmonic Analysis: Real-variable Methods, Orthogonality and Oscillatory Integrals. Princeton University Press. ISBN 0-691-03216-5.^[8]
• Stein, Elias; Shakarchi, R. (2003). Fourier Analysis: An Introduction. Princeton University Press. ISBN 0-691-11384-X. 2011 reprint
• Stein, Elias; Shakarchi, R. (2003). Complex Analysis. Princeton University Press. ISBN 0-691-11385-8. 2010 reprint
• Stein, Elias; Shakarchi, R. (2005). Real Analysis: Measure Theory, Integration, and Hilbert Spaces. Princeton University Press. ISBN 0-691-11386-6. 2009 reprint
• Stein, Elias; Shakarchi, R. (2011). Functional Analysis: An Introduction to Further Topics in Analysis. Princeton University Press. ISBN 978-0-691-11387-6.

• This article incorporates material from Elias Stein on PlanetMath, which is licensed under the Creative Commons Attribution/Share-Alike License.

• Elias M. Stein at the Mathematics Genealogy Project
• Citation for Elias Stein for the 2002 Steele prize for lifetime achievement
• Elias Stein Curriculum Vitae
• Fefferman, Charles; Ionescu, Alex; Tao, Terence; Wainger, Stephen (October 2020). "Analysis and applications: The mathematical work of Elias Stein". Bulletin of the American Mathematical Society. 57 (4): 523–594. doi:10.1090/bull/1691.
• "Simons Foundation: Elias Stein". Simons Foundation. 2017-03-13. – Extended video interview.