**Cesare Arzelà** was an Italian mathematician best known for his work in the theory of functions of a real variable.

- We note that this was a period of transition for education in Italy, with the newly unified country taking over the regulation of colleges and many changes occurred during Arzelà's four years of study.
- Betti was also director of its teacher's college which Arzelà was attending, the Scuola Normale Superiore, from 1864.
- In 1869 Arzelà graduated having completed work on his dissertation on potential theory.
- After graduating, Arzelà continued to attend courses in higher analysis, mathematical physics, and higher mechanics while continuing to work for the certificate from the Scuola Normale Superiore which would allow him to become a secondary school teacher.
- Leaving Pisa after studying there for a year, Arzelà taught in Savona.
- There were two students that Arzelà taught at the Instituto Tecnico in Florence who turned out to play an important role in his future.
- These were Rodolfo Bettazzi (1867-1941) and Vito Volterra - both became university teachers (in Volterra's case becoming a mathematician of top international importance) and went from being Arzelà's students to becoming colleagues and friends.
- Arzelà came out on top and took up the position in 1878.
- During the years that Arzelà had been working his way towards a university post he had been making use of his teaching experiences writing a textbook.
- Arzelà only taught at Palermo for two years for, having entered a competition for a professorship at the University of Bologna, he was appointed as professor of Infinitesimal Calculus in 1880.
- He proved the result now known as the Ascoli-Arzelà theorem on the existence of a uniformly convergent subsequence in every sequence of equilimited and equicontinuous functions.
- Both Giulio Ascoli and Arzelà had studied the concept of equicontinuity and Arzelà's theorem was a generalisation of a much weaker one which had been proved by Ascoli in 1884.
- These give interesting information about Arzelà's research, and in particular his research in functional analysis which the two mathematicians refer to as "functions of lines" in their correspondence.
- Notice the title of Arzelà's paper Sulle funzioni di linee Ⓣ(On line functions) in which the Ascoli-Arzelà theorem appears.
- Also in this 1895 paper Arzelà expressed the hope that he would be able to apply his results to justify the Dirichlet principle.
- Notice that today the Ascoli-Arzelà theorem is a result about compactness but this idea was only introduced by Maurice Fréchet in 1904.
- Arzelà continued to produce high quality research papers throughout his life.
- This course by Arzelà was the first course on Galois theory to be given in Italy.
- It is interesting to see the material on which Arzelà based his course: Lejeune Dirichlet's Vorlensungen über Zahlentheorie Ⓣ(Lectures on number theory); Joseph Serret's Cours d'algèbre supérieure Ⓣ(A course on advanced algebra), Camille Jordan's Traité des substitutions Ⓣ(Treatise on substitutions) and Eugen Netto's Substitutionentheorie Ⓣ(Substitution theory).
- In the year long course of lectures Arzelà proved that equations of degree greater than four cannot be solved by radicals.
- in 1880, Arzelà had written one of the most widely used secondary school texts.
- Arzelà thus commands an important position in the history of the teaching of mathematics in Italy both at the secondary and at the university level.
- We have mentioned that Salvatore Pincherle was a colleague of Arzelà's at the University of Bologna but we should also mention that Federigo Enriques became a colleague in 1896.
- Among the students that Arzelà taught at Bologna who became leading mathematicians, in addition to Bortolotti who we mentioned above, we should add the names of Leonida Tonelli and Giuseppe Vitali.
- Finally let us mention that Arzelà was elected to the Reale Accademia dei Lincei and, in 1907, shared with Guido Castelnuovo the Royal Prize for Mathematics of 10,000 lire given by the Reale Accademia dei Lincei.

Born 6 March 1847, Santo Stefano di Magra, La Spezia (now Italy). Died 15 March 1912, Santo Stefano di Magra, La Spezia, Italy.

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Origin Italy

**O’Connor, John J; Robertson, Edmund F**: MacTutor History of Mathematics Archive