# Leonida Tonelli

Leonida Tonelli
Born 19 April 1885
Gallipoli, Apulia, Italy
Died 12 March 1946 (aged 60)
Pisa, Italy
Nationality Italian
Fields Mathematics
Institutions University of Cagliari
University of Parma
University of Bologna
University of Pisa
Scuola Normale Superiore
Alma mater University of Bologna
Doctoral students Lamberto Cesari, Guido Stampacchia
Known for Tonelli's theorem

Leonida Tonelli (19 April 1885 – 12 March 1946) was an Italian mathematician, noted for creating Tonelli's theorem, a variation of Fubini's theorem, and for introducing semicontinuity methods as a common tool for the direct method in the calculus of variations.[1]

## Education

Tonelli graduated from the University of Bologna in 1907.

## Work

He is one of the founders of Modern Theory of Functions of Real Variables and his work on the Calculus of Variations is a milestone in analysis.[2]
The present writer's father, W. H. Young, used to recall that this very question — what principle can we use as the foundation of the calculus of variations[3] — had been put him by a young Italian mathematician. His reply was a question: "Can you use semicontinuity?" The young Italian was Leonida Tonelli. Semicontinuity was then still a recent concept, known only to a few. In the hands of Tonelli, it became an important tool in a fundamental new approach to the calculus of variations.

## Selected publications

• Opere scelte, a cura dell'Unione matematica italiana e col contributo del Consiglio nazionale delle richerche, 1900
• Fondamenti di Calcolo delle Variazioni. Zanichelli, Bologna, vol. 1: 1922,[4] vol. 2: 1923
• "The Calculus of Variations". Bull. Amer. Math. Soc. 31: 163–172. 1925. doi:10.1090/s0002-9904-1925-04002-1. MR 1561014.
• Serie trigonometriche. Zanichelli, Bologna 1928[5]

## Notes

1. The quote is reported verbatim, including the uppercase emphasis on the mathematical topics cited.
2. As also recalled by Sneddon (1986, p. 36).
3. Dresden, Arnold (1926). "Review: Fondamenti di Calcolo delle Variazioni, Vol. 1, by Leonida Tonelli" (PDF). Bull. Amer. Math. Soc. 32 (4): 381–386. doi:10.1090/s0002-9904-1926-04231-2.
4. Tamarkin, J. D. (1929). "Review: Serie trigonometriche by Leonida Tonelli" (PDF). Bull. Amer. Math. Soc. 35 (6): 871–874. doi:10.1090/s0002-9904-1929-04813-4.