One of the events leading up to World War I was Austria-Hungary making an ultimatum to Serbia in the belief that Russia would be deterred from becoming involved by the threat of German support for Austria. Despite Serbia essentially accepting the ultimatum, Austria-Hungary declared war on Serbia on 28 July 1914. Russia immediately ordered partial mobilization against Austria-Hungary and two days later ordered full mobilization. The Bochner family, fearing a Russian invasion, fled from their home and settled in Berlin. This move had its advantages and disadvantages. On the down side the family was less well off in Berlin, but after only a few months there Salomon passed the entrance examination to a high quality Gymnasium which gave him an outstanding education.
After graduating from the Gymnasium, Bochner entered the University of Berlin in 1918. His PhD, which was supervised by Schmidt (with assistance from Schur), was on orthogonal systems of complex analytic functions. He received his doctorate from the University of Berlin on 13 June 1921. However in the early 1920s Germany faced an escalating rate of inflation that was eventually to destroy the German mark. Bochner's family suffered, as did the majority of Germans, and although he fervently wished to pursue an academic career in mathematics, he felt that he had to support his family through the difficult times. He ran a successful import-export business for a few years but success in business did not bring him happiness as he longed to study mathematics. His family, more interested to see him happy than for the financial benefits his business was bringing them, urged him to return to mathematics despite hyperinflation in Germany in 1923. Since Bergman had been undertaking research on orthogonal systems of analytic functions during the time that Bochner had put his mathematics aside, Bochner decided that he needed to move into a new area.
It was in 1923 that Harald Bohr published his ideas on almost periodic functions and Bochner read these. He extended the ideas on his own, coming up with some original techniques which impressed Harald Bohr who invited him to Copenhagen. In 1924, after the award of an International Education Board fellowship, Bochner travelled to Copenhagen to work with Harald Bohr on almost periodic functions. He was also able to travel to England on the Fellowship where he worked with Hardy in Oxford and Littlewood in Cambridge. In England he began studying the zeta function and this, like almost periodic functions, was to be a continuing interest throughout his life.
Bochner lectured at the University of Munich from 1924 to 1933, submitting an outstanding habilitation dissertation on almost periodic functions in 1927. He developed major results in harmonic analysis :-
... that was perhaps he greatest achievement. The culmination of this work at Munich was the publication in 1932 of 'Vorlesungen über Fouriersche Integrale' , an influential book that has become a mathematical classic. Among much else the book contains Bochner's most famous theorem, characterising the Fourier-Stieltjes transforms of positive measures as positive-definite functions ...This work developed into the theory of distributions, see for example  for details. However the period during which Bochner was producing this outstanding work was also one of great personal difficulty.
The Mathematics Department at the University of Munich, realising what an outstanding mathematician Bochner was, wished to appoint him as an assistant professor but both the University Senate and the local government raised objections. He was, of course, a foreigner, originally an Austrian who had acquired Polish nationality as a consequence of the peace treaties at the end of World War I. The Senate demanded that he become a member of the Reich, but even then they were only prepared to appoint him as a lecturer, and not as an assistant professor. Perron held a mathematics chair at Munich and he argued that the University's reputation was being damaged by what was a political rather than scientific decision. With the assistance of Carathéodory, also a professor at Munich, Perron tried to obtain an invitation for Bochner to go to Harvard:-
... since we could thus have a new means in our hands to make even clearer to the Ministry that Bochner is quite a character and that we make fools of ourselves in front of the scientific world if we put him off life here.The attempt to obtain an invitation for Bochner to Harvard rather backfired, however, since the decision eventually fell to G D Birkhoff, who was visiting Europe. He wrote on 7 July 1928 that it would be better:-
... to get some promising young American, of about equal standing and achievement, to come to us for half a year at a similar salary, ... Bochner's conversation gives me the impression of genuine devotion to his science. He says he is interested in the whole field of analysis. Personally I have heard nothing much of Bochner in Vienna, Budapest, Szeged, Göttingen, or Berlin, which seems to indicate that he is not the man of outstanding promise. It is easy to understand that Carathéodory's recommendations are slightly tinged by his personal relations to Bochner, and also perhaps by a feeling (not rare here) that any second rate European youngster is good enough for us.Poor Bochner was clearly suffering from Birkhoff's anti-Semitism which is now well documented, see . Certainly at this time Birkhoff systematically kept Jews out of his department, and this must be the explanation for the quite ridiculous assessment of Bochner as a "second rate European youngster".
We have mentioned above what is considered to be Bochner's most important work in Munich, but in fact he worked on a surprising variety of topics while he was there. It is certainly worth mentioning that in a 1928 paper on Riemann surfaces he used Zorn's lemma, seven years before Zorn discovered the result independently. He published a generalisation of the Lebesgue integral in 1932, which is now known as the Bochner integral. The breadth of his work, however, is best illustrated by the fact that he was also undertaking research in physics at this time and published several papers on X-ray crystallography.
On 30 January 1933 Hitler came to power and on 1 April there was the so-called "boycott day" when Jewish shops were boycotted and Jewish lecturers were not allowed to enter their university. On 7 April 1933 the Civil Service Law provided the means of removing Jewish teachers from the universities, and of course also to remove those of Jewish descent from other roles. All civil servants who were not of Aryan descent (having one grandparent of the Jewish religion made someone non-Aryan) were to be retired. As an orthodox Jew, Bochner's position was now impossible and he accepted a position at Princeton in 1933.
Appointed as an associate in Princeton in 1933, Bochner was promoted to assistant professor in the following year. Every summer he made a trip to Europe to visit his family who were steadily becoming in greater danger from the Nazi anti-Jewish policies. Bochner persuaded them to leave Germany and go to England. By this time his sister Fanny was married and when the family became convinced that life in Germany was impossible and moved to England, Bochner paid for Fanny's children to attend good schools there. On one of his summer trips to Europe Bochner met Naomi Weinberg who was the daughter of the publisher of a New York Jewish newspaper. On 1 November 1938 he married Naomi with Johnny von Neumann as his best man. They had one daughter, Deborah. He had become a naturalised citizen of the United States in 1938 and, in the following year, he was promoted to associate professor, becoming full professor in 1946. He was appointed as Henry Burchard Fine Professor in 1959 and held this prestigious chair until he retired in 1968.
Zund  summarises his major research contributions in Princeton:-
Bochner's years at Princeton continued to break new ground in startling new directions, including functional analysis and group theory. He collaborated with a number of distinguished mathematicians, perhaps most notably J von Neumann on a theory of almost periodic functions on a group (1935). Bochner found that the Riemann Localisation Theorem was not valid for Fourier series of several variables (1935 - 1936), which led him indirectly to consider functions of several complex variables (1937). He made basic contributions to this theory that included the Bochner-Martinelli Formula (1943), and extensions of Cauchy's integral formula (1944). His research was summarised in Several Complex Variables (1948), written with W T Martin, which was the first American book on the subject.Again showing wide interests, Bochner worked on the theory of probability. His major book on this topic is Harmonic Analysis and the Theory of Probability (1955). He also published papers on the gamma function, the zeta function and partial differential equations. In the 1960's he worked on the history and philosophy of mathematics. One of his most interesting works on this topic is The Role of Mathematics in the Rise of Science (1966). His interests in the philosophy of mathematics led to his becoming one of the editors of the five volume work Dictionary of the History of Ideas (1973-74). He was the only member of the editorial team with a scientific background.
Although he was seventy years old when he retired from Princeton, Bochner was appointed as Edgar Odell Lovett Professor of Mathematics at Rice University and went on to hold this chair until his death in 1982. He became Head of Department at Rice in 1969 and held this position until 1976. Gunning  writes:-
He was as influential at Rice as he had been at Princeton, if not more so ... In addition to teaching mathematics he participated in the history of science program, was responsible for the foundation of an interdisciplinary institute for the history of ideas, and gave university-wide public lectures on the history of science.Bochner received many honours for his outstanding contributions. He was elected to the National Academy of Sciences in 1950. He was American Mathematical Society Colloquium lecturer in 1956, was vice-President of the American Mathematical Society in 1957-58 and was awarded the Society's Leroy P Steele Prize in 1979.
Zund writes :-
Bochner was a personable individual who had an intense curiosity and seemingly boundless energy.
Article by: J J O'Connor and E F Robertson