**Renato Caccioppoli** was an Italian mathematician, known for his contributions to mathematical analysis.

- In the same year, Caccioppoli became Picone's assistant.
- In this work Caccioppoli began to investigate how to generalise Riesz's theorem on the representation of linear functionals by extending the initial definition set.
- In the same year Caccioppoli considered the extension of the definition of linear functionals from the set of continuous functions to the set of Baire functions, anticipating a special case of the Hahn-Banach theorem.
- This approach was later taken up again by Caccioppoli, and it is one of the threads running throughout his work.
- Once again Caccioppoli wanted to apply his "classical" method, that is to say, to extend a functional beyond its initial definition set.
- Caccioppoli then defined the area of a curved surface as the Stieltjes' integral of the area element built with the area elements of the projections of the surface S on coordinate planes.
- Reviews by L C Young initially suggested that Caccioppoli's theory was insufficient in general and only worked in some cases.
- After 1930 Caccioppoli devoted himself to the study of differential equations and he provided existence theorems for both linear and non-linear problems.
- Carrying on in this way Caccioppoli, in 1931, extended in some cases Brouwer's fixed point theorem, and applied his results to existence problems of both partial differential equations and ordinary differential equations.
- In the period between 1933 and 1938 Caccioppoli applied his method to elliptic equations, providing the a priori upper bound for their solutions, in a more general way than Bernstein did for the two-dimensional case.
- Again in 1938 Caccioppoli resumed the study of Riemann's existence theorems, dealing with the existence of abelian integrals on a closed Riemann surface.
- In May 1938 Hitler was visiting Naples with Mussolini: Caccioppoli, who had already shown his opposition to fascism, convinced an open-air restaurant orchestra to play La Marseillaise, and made a speech against Italian and German dictators.
- After the Second World War Renato Caccioppoli resumed his scientific activity.
- In 1952 Caccioppoli sketched a revised vrsion of his early work on surface area and related topics, with the article Misura e integrazione degli insiemi dimensionalmente orientati Ⓣ(Measurement and integration of dimensionally oriented sets), (Rend.
- These finite perimeter sets which were introduced by Caccioppoli are now known as "Caccioppoli sets".
- His last work dates back to 1952-1953 and deals with pseudoanalytic functions - an original concept introduced by Caccioppoli - extending some properties of analytic functions.
- In 1992, a film Morte di un matematica napoletono Ⓣ(Death of a Naples mathematician) was made by the Italian Director Mario Martone about the events leading to Caccioppoli's suicide.
- The Mathematics Department at the University of Naples is named after Renato Caccioppoli.

Born 20 January 1904, Naples, Italy. Died 8 May 1959, Naples, Italy.

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Origin Italy

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