Cesare Arzelà

Cesare Arzelà

Cesare Arzelà (born March 6, 1847 in Santo Stefano di Magra , La Spezia , † March 15, 1912 in Santo Stefano di Magra) was an Italian mathematician.

Arzelà came from a simple background and went to high school in Sarzana from 1856 to 1858 and to the Lyceum of Pisa from 1858 to 1861. From 1861 he studied with a scholarship at the Scuola Normale Superiore in Pisa (with the aim of becoming a teacher) and also at the University of Pisa with the degree in 1869. His teachers were Enrico Betti , who supervised his dissertation on potential theory, and Ulisse Dini . After his teaching diploma in 1870, he initially taught in Macerata , but remained scientifically active and published. In 1872/73 he received permission to continue studying at the University of Pisa. He heard about elasticity theory from Betti and published on deformation of an elastic ellipsoid with application to the earth's shape. He then taught in Savona , from 1875 in Como and then at the Technical Institute in Florence . There he taught his later professor colleagues and friends Rodolfo Bertazzi (1867–1941) and Vito Volterra . From 1878, after a competition, he became professor of algebra in Palermo and, in 1880, professor of analysis at the University of Bologna . His algebra textbook appeared in 1880 and was widely used. In 1884 he received the chair for higher analysis in Bologna. Salvatore Pincherle became a colleague of his in 1881 and among them degrees in mathematics were awarded in Bologna, which was previously not possible because there were no professors for higher mathematics.

He did research in the field of real functions. He worked on the concept of uniform convergence (1883), more precisely he introduced the partially uniform convergence ( called quasi- uniform convergence by Émile Borel in 1905), which, according to him, was a necessary and sufficient condition for the continuity of the limit function against which a sequence of continuous functions converged . 1885 proved a theorem about the interchangeability of Riemann integration with the formation of limit values ​​in the case of Riemann integrable uniformly restricted function sequences ( generalized by Henri Lebesgue in his theorem on majorizing convergence). In 1889 Giulio Ascoli's theorem (1884) was generalized by him to the theorem of Arzelà-Ascoli (published in his article Sulle funzioni di linee 1895). Arzelà-Ascoli's theorem represents an important mathematical theorem in the field of functional analysis and states the existence of a uniformly convergent subsequence for every sequence of equally limited and continuous functions. Later it was taken as a statement about compactness in functional spaces (a concept that Maurice Fréchet introduced in 1904). Arzelà himself hoped to justify the Dirichlet principle with the sentence , but he only succeeded with additional assumptions. He was in correspondence with Volterra. The correspondence is a source for the early phase of functional analysis (both called the theory of funzioni di linee , or line functions). The concept of the line function on sets of curves was influential for the further expansion of functional analysis at Maurice Fréchet. In 1886/87 he gave the first course on Galois theory in Italy, the transcript of which has been preserved. He mainly used a book by Eugen Netto (Substitution Theory) as a suggestion. The impossibility of solving equations with degrees greater than four by radicals, he attributed to Paolo Ruffini , where he possibly had access to manuscripts of Ruffini still existing in Bologna, since the details of his proof were otherwise difficult to access in Arzela's time.

Ettore Bortolotti , Leonida Tonelli and Giuseppe Vitali were among his students .

He was a member of the Accademia dei Lincei and received in 1907 with Guido Castelnuovo their royal prize for mathematics in the amount of 10,000 lire.

Fonts

• Trattato di algebra elementary 1880
• with G. Ingrami: Aritmetica razionale 1894
• Complementi di algebra elementare 1896
• Lezioni di calcolo infinitesimale, 2 volumes, 1901, 1906 (from lectures in Bologna 1880/81)

Web links

• John J. O'Connor, Edmund F. Robertson :  Cesare Arzelà. In: MacTutor History of Mathematics archive .

Individual evidence

1. ↑ Article Cesare Arzela in Guido Walz (ed.), Lexikon der Mathematik, Spektrum Akad. Verlag