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{{Infobox scientist
| name              = Renato Caccioppoli
| image             = caccioppoli.jpg
| image_size        = 
| alt               = 
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| birth_date        = {{Birth date|1904|01|20}}
| birth_place       = [[Napoli]]
| death_date        = {{Death date and age|1959|05|08|1904|01|20}}
| death_place       = [[Napoli]]
| resting_place     = 
| resting_place_coordinates = <!-- {{Coord|LAT|LONG|type:landmark|display=inline,title}} -->
| residence         = 
| citizenship       = 
| nationality       = [[Italia]]n
| fields            = [[Functional analysis]]<br>[[Partial differential equation]]s
| workplaces        = [[Università di Napoli Federico II]]<br>[[Università di Padova]]
| alma_mater        = [[Università di Napoli Federico II]]
| thesis_title      = 
| thesis_url        = 
| thesis_year       = 
| doctoral_advisor  = [[Mauro Picone]]
| academic_advisors = 
| doctoral_students = [[Federico Cafiero]]
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}}
'''Renato Caccioppoli''' ({{IPA-it|reˈnaːto katˈtʃɔppoli|lang}}; 20 January 1904 – 8 May 1959) was an [[Italy|Italian]] [[mathematician]], known for his contributions to [[mathematical analysis]], including the [[Several complex variables|theory of functions of several complex variables]], [[functional analysis]], [[measure theory]].

== Life and career ==
Born in [[Naples]], he was the son of Giuseppe Caccioppoli (1852–1947), a [[surgeon]], and his second wife Sofia Bakunin (1870–1956), daughter of the Russian revolutionary [[Mikhail Bakunin]]. After earning his diploma in 1921, he enrolled in the department of engineering, but in November, 1923 changed to mathematics. Immediately after earning his [[laurea]], in 1925, he became the assistant of [[Mauro Picone]], who in that year was called to the [[University of Naples]], where he remained until 1932. Picone immediately discovered Caccioppoli's gifts and pointed him towards research in [[mathematical analysis]]. In the course of the next five years, Caccioppoli published about thirty works on topics developed in the complete autonomy provided by a ministerial award for mathematics in 1931, a competition he won at the age of 27, the chair of analisi algebrica at the [[University of Padova]]. In 1934 he returned to Naples to accept the chair in [[group theory]]; later he took the chair of superior analysis, and from 1943 onwards, the chair in mathematical analysis.

In 1931 he became a correspondent member of the [[Academy of Physical and Mathematical Sciences of Naples]], becoming an ordinary member in 1938. In 1944 he became an ordinary member of the [[Accademia Pontaniana]], and in 1947 a correspondent member of the [[Accademia dei Lincei]], and a national member in 1958. He was also a correspondent member of the [[Paduan Academy of Sciences, Letters, and Arts]]. In the years from 1947 to 1957 he directed, together with [[Carlo Miranda]], the journal ''[[Giornale di Matematiche]]'', founded by [[Giuseppe Battaglini]]. In 1948 he became a member of the editing committee of ''[[Annali di Matematica]]'', and starting in 1952 he was also a member of the editing committee of ''Ricerche di Matematica''. In 1953 the Academia dei Lincei bestowed on him the national prize of physical, mathematical, and natural sciences.

He was an excellent [[pianist]], noted as well for his nonconformist temperament. He tried out the vagrant life, and was arrested for begging. In May 1938 he gave a speech against [[Adolf Hitler]] and [[Benito Mussolini]], when the latter was visiting Naples. Together with his companion Sara Mancuso, he had the French national [[anthem]] played by an orchestra, after which he began to speak against [[fascism]] and [[Nazism]] in the presence of [[OVRA]] agents. He was again arrested, but his aunt, [[Maria Bakunin]], who at the time was a professor of chemistry at the [[University of Naples]], succeeded in having him released by convincing the authorities that her nephew was ''[[non compos mentis]]''. Thus Caccioppoli was interned, but he continued his studies in mathematics, and playing the piano.

His most important works, out of a total of around eighty publications, relate to [[functional analysis]] and the [[calculus of variations]]. Beginning in 1930 he dedicated himself to the study of [[differential equation]]s, the first to use a topological-functional approach. Proceeding in this way, in 1931 he extended the [[Brouwer fixed point theorem]], applying the results obtained both from ordinary differential equations and partial differential equations.

In 1932 he introduced the general concept of inversion of functional correspondence, showing that a transformation between two [[Banach space]]s is invertible only if it is locally invertible and if the only convergent sequences are the compact ones.

Between 1933 and 1938 he applied his results to [[elliptic equation]]s, establishing the majorizing limits for their solutions, generalizing the two-dimensional case of [[Felix Bernstein]]. At the same time he studied [[analytic function]]s of [[several complex variables]], i.e. analytic functions whose [[domain (mathematics)|domain]] belongs to the vector space '''C'''<sup><math>n</math></sup>, proving in 1933 the fundamental theorem on normal families of such functions: if a family is normal with respect to every complex variable, it is also normal with respect to the set of the variables. He also proved a [[Residue (complex analysis)|logarithmic residue]] formula for functions of two complex variables in 1949.

In 1935 Caccioppoli proved the analyticity of class <math>C^2</math> solutions of elliptic equations with analytic coefficients.

The year 1952 saw the publication of his masterwork on the area of a surface and [[measure theory]], the article ''Measure and integration of dimensionally oriented sets'' (''Misura e integrazione degli insiemi dimensionalmente orientati'', Rendiconti dell'[[Accademia Nazionale dei Lincei]], s. VIII, v.12). The article is mainly concerned with the theory of dimensionally oriented sets; that is, an interpretation of surfaces as oriented boundaries of sets in space. Also in this paper, the family of sets approximable by polygonal domains of finite perimeter, known today as [[Caccioppoli set]]s or [[Caccioppoli set|sets of finite perimeter]], was introduced and studied.

His last works, produced between 1952 and 1953, deal about a class of [[pseudoanalytic function]]s, introduced by him to extend certain properties of [[analytic function]]s.

In his last years, the disappointments of [[politics]] and his wife's desertion, together perhaps with the weakening of his mathematical vein, pushed him into [[alcoholism]]. His growing instability had sharpened his "strangenesses", to the point that the news of his suicide on May 8, 1959 by a gunshot to the head did not surprise those who knew him. He died at his home in Palazzo Cellammare

In 1992 his tormented personality inspired the plot of a film directed by [[Mario Martone]], ''[[The Death of a Neapolitan Mathematician]]'' (''[[Morte di un matematico napoletano]]''), in which he was portrayed by [[Carlo Cecchi]]. He names also an [[asteroid]], [[9934 Caccioppoli]].

==Selected publications==
*{{Citation
| last = Caccioppoli
| first = Renato
| title = Opere scelte 
| place = [[Rome|Roma]]
| publisher = Edizioni Cremonese (distributed by [[Unione Matematica Italiana]])
| year = 1963
| volume =
| url = 
| doi = 
| zbl = 0112.28201
| isbn = 88-7083-505-7 
}} (Volume 1) ISBN 88-7083-506-5 (Volume 2). His "''Selected works'', a selection from Caccioppoli's scientific works with a biography and a commentary.

== See also ==
* [[Contraction principle (large deviations theory)|Contraction principle]]
* [[Caccioppoli set]]
* [[Weyl's inequality]]

==Biographical references ==
This article is based largely on material from [[:it:Renato Caccioppoli|the equivalent article on Italian Wikipedia]], accessed 4 March 2006, and also on the following biographical works:
*{{Citation
| last = Fichera
| first = Gaetano
| author-link = Gaetano Fichera
| title = Ricordo di Renato Caccioppoli
| language = Italian
| journal = [[Ricerche di Matematica]]
| volume = 40
| issue = supplement
| pages = 11–15
| year = 1991
| zbl = 0788.01051
}}. The "''Recollections of Renato Caccioppoli''" (English translation of the title) by one of his [[colleague]]s and close friend.
*{{Citation
 | last = Sbordone
 | first = Carlo
 | author-link = Carlo Sbordone
 | title = Renato Caccioppoli, nel centenario della nascita
 | journal = Bollettino della [[Unione Matematica Italiana]], Sezione A, La Matematica nella Società e nella Cultura
 | series = Serie VIII
 | volume = 7
 | issue = 2
 | pages = 193–214
 | year = 2004
 | language = Italian
 | url = http://mathematica.sns.it/media/volumi/214/Caccioppoli.pdf
 | doi =
 | id =
 | mr = 2097985
 | zbl =1192.01026
}}. "''Renato Caccioppoli, on the centenary of his birth''" is an ample biographical paper by Carlo Sbordone, pupil of Federico Cafifiero.
*{{Citation
  | last = UMI
  | author-link = Unione Matematica Italiana
  | title = Renato Caccioppoli
  | language = Italian
  | journal = [[Bollettino dell'Unione Matematica Italiana]]
  | series = Serie III
  | volume = 14
  | issue = 2
  | pages = 294
  | year = 1959
  | url = http://www.bdim.eu/item?id=BUMI_1959_3_14_2_264_0
  | id =
  | mr =
  | zbl =
}}. A brief obituary, basically announcing the commemoration of his scientific work published in the following issue 4 of the same Bulletin.
*{{Citation
  | last = UMI
  | author-link = Unione Matematica Italiana
  | title = L'opera matematica di Renato Caccioppoli
  | language = Italian
  | journal = [[Bollettino dell'Unione Matematica Italiana]]
  | series = Serie III
  | volume = 14
  | issue = 4
  | pages = 548–551
  | year = 1959
  | url = http://www.bdim.eu/item?id=BUMI_1959_3_14_4_548_0
  | id =
  | mr =
  | zbl =
}}. "''The mathematical work of Renato Caccioppoli''", a relation on his research work published in the UMI Bulletin: even if no author is stated, {{harvtxt|Sbordone|2004|loc=reference [21], p. 212}} attributes the article to [[Gianfranco Cimmino]].

== References ==
*{{Citation
 | first =
 | last =
 | author-link =
 | first2 =
 | last2 =
 | author2-link =
 | editor-last = Alvino
 | editor-first = A.
 | editor-link =
 | editor2-last = Carbone
 | editor2-first = L.
 | editor2-link =
 | editor3-last = Sbordone
 | editor3-first = C.
 | editor3-link = Carlo Sbordone
 | editor4-last = Trombetti
 | editor4-first = G.
 | editor4-link =
 | contribution =
 | contribution-url =
 | title = In ricordo di Renato Caccioppoli
 | place = Napoli
 | publisher = Giannini
 | origyear = 1999
 | year = 2004
 | edition = 2nd printing
 | pages = 124
 | language = Italian
 | url =
 | doi =
 | id =
 | isbn =
 | mr =
 | zbl = 0928.00071
}}. "''In memoriam Renato Caccioppoli''": a collection of papers detailing his personality and his research, including the introduction to his "''Opere scelte''" (Selected works), a list of contributions from the "''International Symposium Renato Caccioppoli''" held in Napoli on September 20–22, 1989, a conference held by Caccioppoli himself and related letters by [[Carlo Miranda]], Giovanni Prodi and [[Francesco Severi]].
*{{Citation
| last = Cafiero
| first = Federico
| author-link = 
| title = Funzioni additive d'insieme e integrazione negli spazi astratti
| place = [[Napoli]]
| publisher = [[Liguori Editore|Libreria Editrice Liguori]]
| year = 1953
| pages = 178
| id = 
| mr = 0215954 
| zbl = 0050.27801
| language = Italian
}}. "''Additive set functions and integration in abstract spaces''" (Italian translation of the title) is the prize winning first monograph where Cafiero states and proves his convergence theorem.
*{{Citation
| last = Cafiero
| first = Federico
| author-link = 
| title = Misura e integrazione
| place = [[Rome|Roma]]
| publisher = Edizioni Cremonese
| year = 1959
| series = Monografie matematiche del [[Consiglio Nazionale delle Ricerche]]
| volume = 5
| pages = VII+451
| id = 
| mr = 0215954 
| zbl = 0171.01503
| language = Italian
}}. ''Measure and integration'' (English translation of the title) is a definitive monograph on integration and measure theory: the treatment of the limiting behavior of the integral of various kind of [[sequences]] of measure-related structures (measurable functions, [[measurable set]]s, measures and their combinations) is somewhat conclusive.
*{{Citation
| last = Cesari
| first = Lamberto
| author-link = Lamberto Cesari
| title = Surface Area
| place = [[Princeton, New Jersey]]
| publisher = [[Princeton University Press]]
| year = 1956
| series = [[Annals of Mathematics Studies]]
| volume = 35
| edition = 
| url = http://books.google.it/books?id=JN3U6cBoS3QC&printsec=frontcover&hl=it#v=onepage&q&f=true
| doi = 
| pages = x+595
| mr = 0074500
| zbl = 0073.04101
| isbn = 0-691-09585-X}}. His work summarizing the theory of surface area, including his own contributions.
*{{Citation
 | last = Miranda
 | first = Carlo
 | author-link = Carlo Miranda
 | title = Equazioni alle derivate parziali di tipo ellittico
 | place = Berlin – [[Göttingen]] – New York
 | publisher = [[Springer Verlag]]
 | series = [[Ergebnisse der Mathematik und ihrer Grenzgebiete]] – Neue Folge
 | volume = Heft 2
 | year = 1955
 | edition = 1st
 | pages = VIII+222
 | language = Italian
 | url =
 | doi =
 | id =
 | isbn =
 | mr = 0087853
 | zbl = 0065.08503
}}.
*{{Citation
 | last = Miranda
 | first = Carlo
 | author-link = Carlo Miranda
 | title = Partial Differential Equations of Elliptic Type
 | place = Berlin – Heidelberg – New York
 | publisher = [[Springer Verlag]]
 | series = [[Ergebnisse der Mathematik und ihrer Grenzgebiete]] – 2 Folge
 | volume = Band 2
 | origyear = 1955
 | year = 1970
 | edition = 2nd Revised
 | pages = XII+370
 | url =
 | doi =
 | id =
 | isbn = 978-3-540-04804-6
 | mr = 0284700
 | zbl = 0198.14101
}}, translated from the Italian by Zane C. Motteler.

== External links ==

* {{MacTutor|last=Faber Fabbris|first=Renato|id=Caccioppoli|title=Renato Caccioppoli|date=January 2000}}
* {{Citation
  | last = The Caccioppoli Family
  | first = 
  | last2 = 
  | first2 = 
  | title = Renato Caccioppoli
  | language = [[English language|English]] and [[Italian language|Italian]]
  | date = July 1, 1997
  | url = http://caccioppoli.com/Renato%20Caccioppoli.htm 
  | accessdate = April 9, 2011}}: biographical sketch from the [http://caccioppoli.com/index.htm Caccioppoli family web site].

{{Authority control|VIAF=54220339}}

{{Persondata <!-- Metadata: see [[Wikipedia:Persondata]]. -->
| NAME              = Caccioppoli, Renato
| ALTERNATIVE NAMES =
| SHORT DESCRIPTION = Italian mathematician
| DATE OF BIRTH     = 20 January 1904
| PLACE OF BIRTH    = Napoli
| DATE OF DEATH     = 8 May 1959
| PLACE OF DEATH    = Napoli
}}
{{DEFAULTSORT:Caccioppoli, Renato}}
[[Category:1904 births]]
[[Category:1959 deaths]]
[[Category:People from Naples]]
[[Category:20th-century Italian mathematicians]]
[[Category:Italian people of Russian descent]]
[[Category:Functional analysts]]
[[Category:Measure theorists]]
[[Category:Mathematical analysts]]
[[Category:Mathematicians who committed suicide]]
[[Category:PDE theorists]]
[[Category:Suicides by firearm in Italy]]
[[Category:University of Naples Federico II alumni]]
[[Category:Italian atheists]]