Hellinger entered the University of Heidelberg but, following the tradition in Germany at that time, he did not complete his studies at a single university but moved to several different universities during the course of his studies. His second university was Breslau, so he returned to the town where he was brought up, and then in 1904 he followed his friend Max Born to Göttingen. Hellinger would keep in touch with Born and developments in quantum mechanics for much of his life. In Göttingen Hellinger was a student of Hilbert and, not long after he began his studies there, he was joined by Courant and Toeplitz who had been his fellow students at Breslau.
Hellinger was awarded his doctorate by the University of Göttingen in 1907 for a thesis entitled Die Orthogonalinvarianten quadratischer Formen von unendlich vielen Variablen Ⓣ. He introduced a new type of integral, the Hellinger integral in his doctoral thesis and, jointly with Hilbert, he produced the important Hilbert-Hellinger theory of forms. Then from 1907 to 1909 he was an assistant at Göttingen and, during this time, he :-
... edited Hilbert's lecture notes and Felix Klein's influential Elementarmathematik vom höheren Standpunkte aus Ⓣ (Berlin, 1925) which was translated into English (New York, 1932).From Göttingen Hellinger went to Marburg where he was a Privatdozent from 1909 to 1914. He was then appointed to a chair at the new university of Frankfurt am Main. He was not the only appointment to Frankfurt in 1914 for Szász was appointed a Privatdozent in that year, later to be promoted to professor. Bieberbach was also at Frankfurt in this early period, although he left after World War I to take up a chair at Berlin. Of course 1914 marked the start of World War I and Hellinger was involved in war service. However, after the end of the war, there were a number of further important appointments to Frankfurt which built up an impressive mathematics department there. Epstein was appointed in 1919, Dehn in 1921 and Siegel in 1922. Others such as Toeplitz were frequent visitors to the Frankfurt Mathematics Seminar. This Seminar is described in  and  which both concentrate on the period from 1922 to the difficult years of the 1930s.
On 30 January 1933 Hitler came to power and 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. However, there was an exemption clause which exempted non-Aryans who had fought for Germany in World War I. Hellinger certainly qualified under this clause and this allowed him to keep his lecturing post in Frankfurt in 1933.
Hellinger, however, was forced to retire in 1936 because by this stage the rules that non-Aryans who served in World War I were allowed to keep their posts was being ignored after decisions at the Nuremberg party congress in the autumn of 1935. Hellinger continued to live in Frankfurt. On the Kristallnacht (so called because of the broken glass in the streets on the following morning), the 9-10 November 1938, 91 Jews were murdered, hundreds were seriously injured, and thousands were subjected to horrifying experiences. Thousands of Jewish businesses were burnt down together with over 150 synagogues. The Gestapo arrested 30,000 well-off Jews and a condition of their release was that they emigrate. The Gestapo did not arrest Hellinger that night because there was nowhere else to put prisoners but :-
he refused to flee ... because he wanted to stay and see just how far beyond the traditional standards of justice and ethics the authorities would go in his case.On 13 November 1938 he found out how far the authorities would go. He was arrested, first taken to the Festhalle and then put into Dachau concentration camp. By this time his sister, Hanna Meissner, was in the United States and she describes in  how Siegel wrote to her to say that Hellinger had been sent to the concentration camp. Fortunately friends were able to arrange a temporary job for Hellinger at Northwestern University at Evanston in the United States. He was released from the Dachau camp after six weeks on condition that he emigrate immediately. Siegel writes :-
I saw Hellinger in Frankfurt a few days after his release. He looked emaciated from the utterly insufficient diet at the camp, but maintained a strong will to live as a result of his impending emigration. He refused to discuss his horrifying experiences and was never able to forget the humiliation that was done him.He emigrated to the United States in late February 1939.
Hellinger's position at Evanston throughout the war was precarious with a series of one-year appointments but he acquired American citizenship in 1944 and worked at Evanston until 1949 when he retired. Of course after just a few years work he would not have received a pension to enable him to live and he had constant financial worries.After retirement he accepted a post at Illinois Institute of Technology but he fell ill with cancer in November 1949 and died a few months later.
At Frankfurt, Hellinger had continued his mathematical work on the spectral theory of Jacobi forms and continued fractions. He did important work on Stieltjes's moment problem. With Toeplitz he wrote a monumental survey of the literature on integral equations up to 1923 for Klein's Enzyklopädie der Mathematischen Wissenschaften. This article, Integralgleichungen und Gleichungen mit unendlich vielen Unbekannten Ⓣ, was published in the Encyclopedia in 1927 and then as a separate article the following year. It is now considered one of the great classics and has been reprinted several times. Hellinger also worked on the history of mathematics and, while at Frankfurt, he wrote an important paper on James Gregory.
Hellinger was a gifted teacher who concerned himself deeply with the students who he taught :-
He always took the welfare of the students to heart even outside his lectures and study groups.The letter  is written by two of Hellinger's students and praises highly his teaching prowess:-
At Northwestern he was one of the most liked professors in the Mathematics Department and was recognised by all for the excellence, skill, and understandability of his lectures and especially his proofs.He had an odd sense of humour but a strong sense of loyalty and some equally strong dislikes. He had a gift for making friends. In  a story shows his sense of humour:-
Shortly after his arrival at Northwestern, one of the professors in describing Northwest's mathematics program to him remarked that in the honours course Felix Klein's 'Elementary mathematics from an advanced standpoint' was used as a text and "perhaps Hellinger was familiar with it". At this Hellinger ... replied "familiar with it, I wrote it!".
Article by: J J O'Connor and E F Robertson