Giuseppe Vitali: Real and Complex Analysis and Differential Geometry

Giuseppe Vitali’s mathematical output has been analysed from various points of view: his contributions to real analysis, celebrated for their importance in the development of the discipline, were accompanied by a more correct evaluation of his works of complex analysis and differential geometry, which required greater historical investigation since the language and themes of those research works underwent various successive developments, whereas the works on real analysis maintained a collocation within the classical exposition of that theory. This article explores the figure of Vitali and his mathematical research through the aforementioned contributions and, in particular, the edition of memoirs and correspondence promoted by the Unione Matematica Italiana, which initiated and encouraged the analysis of his scientific biography. Vitali’s most significant output took place in the first 8 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 non-measurable 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). Vitali’s academic career was interrupted by his employment in the secondary school and by his political and trade union commitments in the National Federation of Teachers of Mathematics (Federazione Nazionale Insegnanti di Matematica, FNISM), which brought about a reduction, and eventually a pause, in his publications. Vitali took up his research work again with renewed vigour during the national competition for university chairs and then during his academic activity firstly at the University of Modena, then Padua and finally Bologna. In this second period, besides significant improvements to his research of the first years, his mathematical output focussed on the field of differential geometry, a discipline which in Italy was long renowned for its studies, and particularly on some leading sectors like connection spaces, absolute calculus and parallelism, projective differential geometry, and geometry of the Hilbertian space. Vitali’s connection with Bologna was at first related to his origins and formation. Born in Ravenna, Vitali spent the first 2 years of university at Bologna where he was taught by Federigo Enriques and Cesare Arzelà. Enriques commissioned him with his first publication, an article for the volume Questioni riguardanti la geometria elementare on the postulate of continuity. Vitali then received a scholarship for the Scuola Normale Superiore and later completed his university studies at Pisa under the guidance of Ulisse Dini and Luigi Bianchi. From the end of 1902 until the end of 1904, when he was teaching in a secondary school in Voghera, Vitali returned to mix with the Bologna circles as he, at times, resided there. During his last years, after spending some time at the Universities of Modena and Padua, Vitali returned to teach at Bologna University, a role he carried out with energy and generosity but unfortunately not long enough to be able to found a school.