**Giuseppe Vitali**was the eldest of his parent's five children; the four younger children were Vitichindo, Goffredo, Sara, and Maria Lodomilla. His father, Domenico Vitali, worked for the railway company while his mother, Zenobia Casadio, was at home looking after the children. Giuseppe's elementary school education, which he completed in 1886, was in Ravenna. He then spent three years at the Ginnasio Comunale in Ravenna where his performance in the final examinations of 1889 was far from outstanding. He continued his secondary education in Ravenna at the Dante Alighieri High School. There he was taught mathematics by Giuseppe Nonni who quickly realised the great potential of his young pupil and advised Vitali's father to allow his son to pursue studies in mathematics. Nonni wrote on 28 June 1895 [2]:-

After graduating from the Dante Alighieri High School, Vitali studied for two years at the University of Bologna, beginning in the autumn of 1895. His main teachers at Bologna were Cesare Arzelà, who had held the chair of Higher Analysis, and Federigo Enriques who taught at Bologna on a temporary basis from 1894, but held the chair of projective and descriptive geometry from 1896. Both Arzelà and Enriques were impressed by their young student and supported his application for a scholarship to study at the Scuola Normale Superiore in Pisa.Your son Giuseppe, who has been my student for five years in the classical school of Ravenna, has always shown great attitude and lots of love for the study of mathematics, so I harbour the highest hopes for him, if he may be given the opportunity to continue to higher studies. His marks, combined with his goodness and modesty, make him the best student I have had so far.

Vitali was awarded the scholarship and began his studies at Pisa in the autumn of 1897. There he was strongly influenced by Luigi Bianchi, who taught him analytic geometry, and Ulisse Dini who taught him infinitesimal calculus. His other lecturers included Eugenio Bertini, Gian Antonio Maggi and Cesare Finzi. He graduated from the Scuola Normale Superiore in Pisa on 3 July 1899 having written the thesis *Intorno alle funzioni analitiche sulle superficie di Riemann* Ⓣ . His thesis advisor had been Bianchi. Vitali published three papers in 1900; two of them (*Sulle funzioni analitiche sopra le superficie di Riemann* Ⓣ and *Sui limiti per n = infinity delle derivate n ^{me} delle funzioni analitiche* Ⓣ) were on the material that he had produced for his thesis, while the other (

*Sulle applicazioni del Postulato della continuità nella geometria elementare*Ⓣ) was an essay on the applications of the postulate of continuity in elementary geometry. From November 1899, he assisted Ulisse Dini for two years while at the same time he studied for a teaching diploma, again advised by Bianchi. His thesis,

*Le equazioni di Appell del 2*Ⓣ was published in 1902 in a shortened version and as a full version in the following year.

^{o}ordine e le loro equazioni integrali
After the award of his teaching diploma, Vitali left university level mathematics to become a secondary school teacher. His move away from university mathematics was probably due to financial problems; secondary school teachers were more highly paid than assistants at a university. He taught first at Sassi, then at Voghera. From 1904 until 1923 he taught at the Liceo C Colombo in Genoa where he became involved in politics becoming a Socialist councillor. Maria Teresa Borgato writes [4]:-

Vitali did enter competitions for chairs, for example in October 1910 he competed for the chair of algebraic analysis at Parma. The referees were Salvatore Pincherle, Luigi Bianchi, Gregorio Ricci-Curbastro, Luigi Berzolari and Onorato Nicoletti who, although they did not recommend Vitali to be appointed, gave this assessment of his contributions in their report:-At that time it was not unusual for a person to begin a university career with a period of secondary school teaching(other examples are Luigi Cremona and Cesare Arzelà), but it was a great pity that it took the university up till1922to concede full research activity to such talent like that of Vitali. Vitali, however, kept in contact with his mentors, and Arzelà in particular remained a constant point of reference for him, through exchange of letters and also meetings above all during the holiday periods(Vitali returned to Bologna during his teaching years in Voghera, and Arzelà, in his turn, spent long periods in Santo Stefano Magra in Liguria). He always maintained a friendship with Enriques, who had followed his transfer from the University of Bologna to the Scuola Normale Superiore of Pisa, but his lifelong friend was Guido Fubini, who had been a fellow student in Pisa.

He did receive offers of temporary positions, such as one from Mario Pieri to teach infinitesimal analysis in Parma in 1917-18. However, despite his obvious research talents, he continued as a secondary school teacher. When the Fascists came to power in 1922, they dissolved the Socialist Party. His political career at an end, Vitali returned to mathematics but he seems to have anticipated this since he had published six papers in the two years 1921 and 1922.The works of[Vitali]show his through understanding of the various branches of analysis, his acumen in the treatment of questions that are delicate and not easy, arriving at interesting results, remarkable clarity and sobriety of composition. The fact itself of his having met with one of the creators of the new directions of integral calculus, Lebesgue, is proof that he stayed within the mainstream of this research, in which such results naturally presented themselves.

First Vitali was appointed to a chair in Modena. This was as a result of a competition for the chair of infinitesimal analysis in the University of Modena in 1923. The referees were Guido Fubini, Tullio Levi-Civita, Salvatore Pincherle, Leonida Tonelli and Gabriele Torelli. In fact Vitali was only ranked as second choice by the referees (although Fubini and Levi-Civita had ranked him top), but the candidate who came top, Gustavo Sannia, did not accept the post when offered it. Then, in December 1925, Vitali was appointed to the chair of mathematical analysis at the University of Padua. Despite serious health problems, Vitali was able to make huge contributions to Padua during the five years that he worked there. He became the first director of the 'Seminario matematico of the University of Padua' which he himself founded. This aimed to encourage both teaching and research in mathematics in Padua by organising conferences, workshops and seminars. Vitali was keen to set up a journal as part of the work of the Seminario and the *Rendiconti del Seminario Matematico dell'Università di Padova* began publishing in 1930. He published the paper *Determinazione della superficie di area minima nello spazio hilbertiano* Ⓣ in the first volume of this journal which continues to the present day (only failing to publish in 1944 and 1945 due to World War II). Vitali published a remarkable volume of mathematics over these years and was an invited speaker at the International Congress of Mathematicians held in Bologna in September 1928, giving the lecture *Rapporti inattesi su alcuni rami della matematica* Ⓣ.

In 1930 Vitali moved to the chair of mathematics at the University of Bologna. Here is an extract from his inaugural lecture at Bologna, given on 4 December 1930 (see [10]):-

Viola writes about his teacher, fifty years after Vitali died (see [14]):-To the young people who dedicate themselves to the study of mathematics, I say that the ability required to find the most suitable and most elegant devices is acquired through practise, through the examination of many examples, through the effort to imitate. To imitate, but not too much, if our discipline is not to become a marsh, a large one to be sure, but stagnant, with neither life nor movement. First imitate to learn, and then renew ourselves. ... I love, at least when I am able, to regard science from a personal point of view, and always, again when possible, go beyond current opinions and look at the problem from a new perspective. I have the impression that some ways must be left behind, some mental habits must be abandoned, if we are not to clip the wings of progress. Even to science we must sometimes repeat Charon's cry: By another way, by other ports, not here, you will find passage across the shore. In my role as teacher I hope to be able to show you other ways, if not other ports.

His significant mathematical discoveries include a theorem on set-covering, the notion of an absolutely continuous function and a criteria for the closure of a system of orthogonal functions. Since he worked very much on his own, his work involves some rediscovering of known results but also some remarkably original discoveries. Maria Teresa Borgato gives this summary of his contributions [4]:-I had just graduated a few months before in pure mathematics, and was a volunteer assistant in the course of the theory of functions. Vitali was at the height of his scientific career, both in terms of the recognition he received - alas, so tardily! - by the highest Italian academies, and for having begun and, in part, concluded the preparation of his treatises, as well as for having begun, with renewed, surprising creative capacities, that research in stellar astronomy ... Vitali was extraordinarily generous and good to me. Beginning with my degree thesis, at that time I was publishing my first works on functions of a real variable with a unilateral derivative. He came to see me, spoke with me at length, he presented me with extracts of his most beautiful, most famous publications on set theory, and on functions of a real variable, he gave me much good advice, he made it so I opened up to him about the whole of my studies and my projects for the future, and that I kept him informed about the results - alas, so modest! - of my research.

One of the themes which ran through Vitali's research was his interest in the ideas of Lebesgue - we have seen many references to this above. We note one further work in this area, namely his paperVitali's most significant output took place in the first eight years of the twentieth century when Lebesgue's measure and integration were revolutionising the principles of the theory of functions of real variables. This period saw the emergence of some of his most important general and profound results in that field: the theorem on discontinuity points of Riemann integrable functions(1903), the theorem of the quasi-continuity of measurable functions(1905), the first example of a nonmeasurable set for Lebesgue measure(1905), the characterisation of absolutely continuous functions as antidervatives of Lebesgue integrable functions(1905), the covering theorem(1908). In the complex field, Vitali managed to establish fundamental topological properties for the functional spaces of holomorphic functions, among which the theorem of compacity of a family of holomorphic functions(1903-1904).

*Sulla definizione di integrale delle funzioni di una variabile*Ⓣ(1925). An English tanslation of the paper was published in 1997. We quote Vitali's introduction from that translation:-

From 1926 Vitali developed a serious illness and, with a paralysed arm, he could no longer write. Nevertheless about half his research papers were written in the last four years of his life after the illness struck. In his last years he worked on a new absolute differential calculus and a geometry of Hilbert spaces. These topics were not followed up by later mathematicians.In a recent work, Professor Beppo Levi introduces for didactic purposes a new definition of integral for bounded functions. He writes: "For measurable functions the notion of integral we introduce coincides with the Lebesgue integral: however, it might be the case that this notion may be applied to nonmeasurable functions". I prove that the definition of Beppo Levi is actually altogether equivalent to that of Lebesgue.

Viola recalled the moment when he was in Paris and he heard of Vitali's death (see [14]):-

After his death, his workThe impression on my mind made by the announcement of his death can never be erased, an announcement that reached me in Paris, during a seminar I was taking part in. I heard people whispering "Vitali est mort!" A few days later I received the painful confirmation directly from Bologna: the Maestro had fallen, struck down suddenly while walking arm in arm with his colleague Ettore Bortolotti, under the porticos of that learned city in which I had spent the most beautiful years of the university studies, and in which were laid, and lie still, the remains of those who gave me life.

*Moderna teoria delle funzoni d variabile reale*Ⓣ was completed and published in 1935.

Vitali was honoured with election to the Academy of Sciences of Turin in 1928, to the Accademia Nazionale dei Lincei in 1930, and to the Academy of Bologna in 1931.

**Article by:** *J J O'Connor* and *E F Robertson*

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