(1885-1946)
He began studying engineering in Bologna in 1902, but then moved to mathematics. His teachers were Arzelà and Pincherle as well as Enriques and Donati. He graduated under Arzelà's supervision in 1907 and became his assistant. Already in 1908 he published his works on continuous and rectifiable parametric curves, i.e. of finite length according to Jordan. These results are still definitive today. However, his major contribution is a direct method for an existential theory for the absolute minimum and maximum of functionals of the calculus of variations. Tonelli also managed to transfer the concept of semicontinuity introduced by Baire from real analysis to functional calculus. He obtained existence theorems containing only qualitative hypotheses for which it is immediate to recognize whether a calculus of variations problem admits an absolute maximum or minimum in a given class, without further analysis.
In 1922 the University of Bologna called him to the chair of Higher Analysis which had already been occupied by his teachers. In that period he also received many awards at an international level. In 1926 he studied the Lebesgue area problem of continuous nonparametric surfaces and arrived at results formally similar to those he had obtained for the length of continuous curves. He then applied the new concepts to the formulation of his direct method for the calculus of variations for multiple integrals (1932). He also worked on functions of real variables, applied mathematics, differential equations, and trigonometric Fourier series. He left Bologna to go to Pisa in 1931, after having published the influential text "Serie Trigonometriche" (Trigonometric Series, 1926).
(Source: "Il Dipartimento di Matematica dell'Università di Bologna: Personale, strutture, attività di ricerca - Anno accademico 1988-89" a cura di M. Bernabei e P. Negrini, editrice CLUEB)