# Ulisse Dini

**Ulisse Dini** (14 November 1845 – 28 October 1918) was an Italian mathematician and politician, born in Pisa. He is known for his contribution to real analysis, partly collected in his book "*Fondamenti per la teorica delle funzioni di variabili reali*".^{[1]}

Dini attended the Scuola Normale Superiore in order to become a teacher. One of his professors was Enrico Betti. In 1865, a scholarship enabled him to visit Paris, where he studied under Charles Hermite as well as Joseph Bertrand, and published several papers. In 1866, he was appointed to the University of Pisa, where he taught algebra and geodesy. In 1871, he succeeded Betti as professor for analysis and geometry. From 1888 until 1890, Dini was *rettore*^{[2]} of the Pisa University, and of the *Scuola Normale Superiore* from 1908 until his death in 1918.

He was also active as a politician: in 1871 he was voted into the Pisa city council, and in 1880, he became a member of the Italian parliament.

He has been elected honorary member of London Mathematical Society.^{[3]}

Thus, by the year 1877, or seven years from the time he began, he published the treatise, since famous, entitled Foundations for the Theory of Functions of Real Variables (Fondamenti per la teoria delle funzioni di variabili reali). Much of what Dini here sets forth concerning such topics as continuous and discontinuous functions, the derivative and the conditions for its existence, series, definite integrals, the properties of the incremental ratio, etc., was entirely original with himself and has since come to be regarded everywhere as basal in the real variable theory.

Nell'analisi del XX secolo ha avuto innanzitutto ampio sviluppo la teoria delle funzioni di variabili reali (inaugurata nel 1878 da un libro del Dini) in relazione alle operazioni classiche del calcolo.^{[4]}

Dini worked in the field of mathematical analysis during a time when it was begun to be based on rigorous foundations. In addition to his books, he wrote about sixty papers.^{[5]}

He proved the Dini criterion for the convergence of Fourier series and investigated the potential theory and differential geometry of surfaces, based on work by Eugenio Beltrami.

His work on the theory of real functions was also important in the development of the concept of the measure on a set.^{[6]}

The implicit function theorem is known in Italy as the Dini's theorem.