**Salvatore Pincherle** was an Italian mathematician who can claim to be one of the founders of functional analysis. He established the Italian Mathematical Union.

- Then Pincherle began his school education at the Lycée Impérial (now the Lycée Thiers) in Marseilles.
- His initial interests were in the humanities, but the school in Marseilles specialised in science teaching and Pincherle soon became fascinated with mathematics through excellent teaching there.
- Betti was a strong influence on Pincherle as was Ulisse Dini who, after working on differential geometry was, by this time, studying and teaching the foundations for the theory of functions of a real variable.
- Pincherle graduated with his laurea in 1874, also earning in the same year his right to lecture, the 'libera docenza', having submitted his two-part thesis Sulle superficie di capillarità Ⓣ(On the capillary surface) and Sulle costanti di capillarità Ⓣ(On the constant of capillarity).
- The quality of his work at the Scuola Normale Superiore of Pisa was very high and it was clear that Pincherle had shown that he had a very promising career in front of him as a university teacher.
- Casorati and Beltrami quickly had Pincherle interested in the new approach to analysis that Bernhard Riemann was taking.
- They encouraged Pincherle to apply for a postgraduate scholarship to enable him to study abroad for a year.
- Let us return to Salvatore Pincherle's biography.
- At Bologna, Pincherle became a colleague of Luigi Donati (1846-1932), who was appointed to Bologna in 1877, and Cesare Arzelà who had been appointed to the chair of Infinitesimal Calculus in 1880.
- Although Pincherle referred to Grassmann and Peano, his approach went far beyond the framework of geometry and placed itself in quite a general context using, in particular, infinite-dimensional linear spaces.
- The reasons for this lack of influence are partly due to the fact that such a general and formal axiomatic approach as proposed by Pincherle did not meet the concerns of most mathematicians in the first years of the 20th century.
- Pincherle defined the notion of dimension and basis, showing that in a set consisting of all linear combinations of nnn independent elements, n+1n+1n+1 elements are always dependent.
- even though he was the author of the article on functional equations and operators in the French version of the "Encyclopédie des mathématique pures et appliquées" (1912), in which he gave a very detailed historical account and referred to his own work, Pincherle's work itself did not have much influence.
- the 1888 paper (in Italian) of S Pincherle on the 'Generalized Hypergeometric Functions' led him to introduce the afterwards named Mellin-Barnes integral to represent the solution of a generalized hypergeometric differential equation investigated by Goursat in 1883.
- Pincherle's priority was explicitly recognized by Mellin and Barnes themselves ...
- wrote: "The idea of employing contour integrals involving gamma functions of the variable in the subject of integration appears to be due to Pincherle, whose suggestive paper was the starting point of the investigations of Mellin (1895) though the type of contour and its use can be traced back to Riemann." In 1910 Mellin ...
- devoted a section (Proof of Theorems of Pincherle) to revisit the original work of Pincherle ...
- In 1915 Pincherle published his lecture notes as Lezioni di Calcolo Infinitesimale Dettata Nella R.
- In addition to his remarkable research contributions and his university teaching, Pincherle was also involved in school level mathematics.
- The Unione Matematica Italiana (Italian Mathematical Union) was established in Bologna by Pincherle on 7 December 1922.
- Pincherle was supported by Luigi Bianchi and Vito Volterra but many other leading Italian mathematicians did not support the idea, being quite happy with regional societies such as the Circolo Matematico di Palermo.
- The fact that the International Mathematical Union was founded in 1920 did much to help Pincherle persuade others for the need for the Unione Matematica Italiana.
- Bologna was accepted for the 1928 International Congress of Mathematicians with Pincherle as president.
- At a meeting of the International Mathematical Union which Pincherle chaired in September 1928, a resolution was unanimously adopted supporting all Pincherle's actions.
- Pincherle received many honours for his contributions.
- Pincherle was awarded the Sacchetti prize by the city of Bologna when he was appointed to the chair there in 1928.
- Pincherle was one of the signatories of the Manifesto but he died before the Fascist regime published its "Manifesto of Race" in 1938 which dismissed Jews from university positions.
- Pincherle's closest friend over the last years of his life had been his colleague Beppo Levi who had signed the "oath to Fascism" in 1931 but was dismissed under the Manifesto of Race.

Born 11 March 1853, Trieste, Austria (now Italy). Died 10 July 1936, Bologna, Italy.

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Algebra, Origin Italy

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