Johann then attended secondary school in Leitmeritz (now Litomerice) in Bohemia between 1897 and 1905. Around half way through his schooling his father retired from the bank in Tetschen and the whole family moved to Leitmeritz. His favourite subjects at this school were mathematics and classical languages. His performance in all subjects was outstanding and for a while he thought that classical languages might he the topic for him to study at university, but later his love of mathematics led him to chose that topic. He did miss the seventh grade at school due to bad health and, in addition to asthma beginning to trouble him at that time, he also had to undergo surgery. His father employed a private tutor to make sure that Johann would be able to enter the eight grade when he had recovered. We have explained how he was attracted by classical languages but he did also consider other directions such as philosophy. He was a talented singer with a beautiful baritone voice and at one stage he also considered training to become an opera singer.
He entered the University of Vienna in 1905 having moved to that city with his parents. The course he studied was a broad one including mathematics, physics, chemistry, logic, philosophy, and he also included some lecture courses on music. In mathematics he took lecture courses by Hans Hahn (one on Theoretical arithmetic and one on the Foundations of geometry), Wilhelm Wirtinger (Ordinary differential equations) and Franz Mertens (one on Algebra and one on Number theory) among others. He was awarded a doctorate in 1910 for a dissertation on the calculus of variations carried out under Gustav von Escherich's supervision. His thesis was examined by Wirtinger and von Escherich.
The winter semester of 1911-12 was spent in Göttingen where he attended lectures by Hilbert, became an assistant at the University of Brünn (now Brno) for a year and then moved to the Technische Hochschule in Vienna to become an assistant to the actuarial mathematician Emanuel Czuber. In 1913 he submitted his dissertation Theory and applications of absolutely additive set functions to the University of Vienna to satisfy the requirements of his habilitation there, and became a privatdozent. Von Escherich examined this habilitation dissertation and wrote:-
While creating a theory of absolutely additive set functions which, heretofore, has barely been investigated, the author succeeds with the development of a theory that contains the theory of integral equations, linear and bilinear forms in infinitely many variables, as a special case. He did this, overcoming considerable obstacles, through a combination of Stieltjes', Lebesgue's and Hellinger's concepts of an integral. The paper is full of original and significant ideas.Radon married Maria Rigele, who was a secondary school teacher of science, in 1916; they had four children the first of which was born in 1917 but only lived 18 days. They then had a son Hermann who was born in 1918, a son Ludwig born in 1919, and a daughter Brigitte born in 1924. Hermann died from an illness in 1939 and Ludwig was killed on the Russian front in 1943. Brigitte went on to obtain a Ph.D. in mathematics and marry the mathematician Erich Bukovics in 1950.
In 1919 Radon became an extraordinary professor at Hamburg shortly after being promoted to an extraordinary professorship in Vienna in an attempt to keep him there. The University of Hamburg was a new university, opening in May 1919 shortly before Radon was appointed on the recommendation of Blaschke. Although happy in Hamburg, he left to become a full professor in Greifswald in 1922 where he succeeded Hausdorff. Radon had an enjoyable social life in Greifswald; he enjoyed sailing and playing the lute.
He moved to Erlangen in 1925 where he filled the chair which had been left vacant when Tietze left to accept the chair in Munich. As in Greifswald, Radon and his wife had an enjoyable social life in Erlangen; they organised fancy dress parties and Radon enjoyed playing the violin in a trio.
In 1928 Radon moved again, this time to the University of Breslau where he succeeded Adolf Kneser on his retirement. After happy times in Greifswald and Erlangen, fate was cruel to Radon in Breslau. His son Hermann was diagnosed as having an incurable disease and, despite strenuous attempts to have him regain his health in Switzerland, he slowly deteriorated. Because of worries with his family, Radon declined the offer of the chair in Vienna in 1938 left vacant when Wirtinger retired. Radon himself became ill and underwent surgery in 1939; he was recovering in hospital when informed of Hermann's death. Of course 1939 marked the beginning of World War II and Radon's son Ludwig served in the German army, dying in 1943 after being fatally wounded on the Russian front. The Germans fortified Breslau but it came under increasing Russian pressure from August 1944 as the Russian offensive swept rapidly west. The centre of Breslau was bombed on 7 October 1944 but the Mathematical Institute was essentially undamaged (only 4 panes of glass were broken). By January 1945 the Russian army was advancing towards Breslau and a decision was taken to move the mathematicians from the city. In February Feigl and his colleagues moved the Mathematical Institute from Breslau to Schönburg Castle at Wechselburg, not far from Leipzig. Radon joined them at Wechselburg where there were no books, lecture notes, or resources of any kind. Radon gave a course on complex analysis completely from memory having no materials available to help him.
Radon's sister-in-law lived in Innsbruck so he made a bold bid to move to that city. He contacted Leopold Vietoris, the only mathematician left there since all the others had been drafted into the military. Receiving an invitation from Vietoris, Radon and his family made the tortuous journey through war-torn Europe and arrived in Innsbruck after many hardships. He started teaching there immediately and remained there until the summer of 1946. After an initial misunderstanding when colleagues in Vienna thought that he was not interested in either of two vacant professorships there, he was appointed to Vienna taking up his appointment on 1 October 1946. He remained there for the rest of his life, serving as dean during 1951-52, and rector in 1954.
Radon applied the calculus of variations to differential geometry which led to applications in number theory. It was while he was studying applications of the calculus of variations to differential geometry that he discovered curves which are now named Radon curves. His best known results involve combining the integration theories of Lebesgue and Stieltjes which first appeared in his habilitation dissertation (which we mentioned above) and then in a second important work Über lineare Funktionaltransformationen und Funktionalgleichungen Ⓣ (1919) :-
It assumed a fundamental importance for functional analysis and has become of equal importance for the application to the logarithmic potential. Next came papers on convex functions and sets and on the determination of functions from the values of their integrals on certain manifolds [Über die Bestimmung von Funktionen durch ihre Integralwerte längs gewisser Mannigfaltigkeiten Ⓣ (1917)] which contains the Radon transform that plays an important role today, especially in medicine and geophysics.During 1918-19 he worked on affine differential geometry, then in 1926 he considered conformal differential geometry. His wide interests led him to study Riemannian geometry and geometrical problems which arose in the study of relativity.
Radon had a long association with the Austrian Academy of Sciences. His doctoral thesis and his habilitation dissertation were both published by the Academy. He was elected a corresponding member of the Academy in 1939 and a full member in 1947. He was chairman of the Mathematical and Physical Section of the Academy from 1952 to 1956. He also served the Austrian Mathematical Society being president in 1948-50.
Article by: J J O'Connor and E F Robertson