At Göttingen Helly studied under Hilbert, Klein, Minkowski and Runge in 1907-8. Returning to Vienna in 1908 he had no university post, but supported himself in a number of different ways. He taught in a Gymnasium, gave private tuition, and wrote solution manuals for a series of standard textbooks. During this period, he undertook research on functional analysis and proved the Hahn-Banach theorem in 1912, fifteen years before Hahn published essentially the same proof and 20 years before Banach gave his new setting. We discuss this more fully later in this article.
On the outbreak of World War I, Helly enlisted in the army. While serving as a lieutenant in September 1915 he was shot. The bullet went through his lung and did damage to his health from which he never recovered throughout the rest of his life. After being shot he was captured by the Russians. After this he spent years in hospitals and prisoner of war camps in Siberia. One might have expected that the end of World War 1 in 1918 would have led to Helly's release but by this time the Russian armies were fighting each other and escape was impossible. Even after leaving Russia it was a long route back to Vienna for Helly who travelled through Japan, the Far East, Egypt and the Middle East before reaching home in 1920.
Helly was married in 1921 to Elise Bloch who also had a doctorate in mathematics from Vienna. In the same year he submitted his habilitation thesis and received the right to teach. He was appointed to Vienna in 1921 but to an unpaid post. His wife believed that he was denied a professorship :-
... partly because Helly was Jewish and also because Hahn thought a younger person should be preferred.Helly was forced to earn a living working in a bank, then as an actuary when the bank collapsed in 1929. He then found a job with an insurance company but even this difficult life became worse in 1938. Hitler himself had entered Austria on 12 March 1938 with the German army, and a Nazi government had been set up there. Helly was dismissed from his post because he was a Jew. He fled from Austria to save himself and his family, emigrating to the United States in 1938.
Life remained difficult for Helly and his family in the United States. At first he resorted to giving private tuition as he had done in Vienna many years before. His first break occurred in 1939 when Einstein supported him for a position in Paterson Junior College in New Jersey. With support from so eminent a person as Einstein, he was successful and received the position. Two years later, in 1941, he moved to Monmouth Junior College, also in New Jersey. The following year saw both Helly and his wife employed as mathematicians by the Signal Corps in Chicago. Helly again took up familiar work since he was preparing mathematics training manuals, while his wife taught mathematics.
It was during his time with the Signal Corps in Chicago that he suffered his first heart attack. This was still as a direct result of the damage which he had suffered when shot during World War I. He recovered from the heart attack and things began to look up when he was offered a chair of mathematics as Illinois Institute of Technology. Sadly he died after a second heart attack shortly after.
He is remembered for Helly's theorem, published in 1923, which states that if there are given n convex subsets of a d-dimensional Euclidean space with n ≥ d+1 and if each collection of d + 1 of the subsets has a point in common then there is a common point of the n subsets.
However, as we noted above, in a paper he published in 1912 there are a number of important results. First there is Helly's selection principle which says that given a sequence of functions of bounded variation which are of uniform bounded variation and uniformly bounded at a point, then there exists a subsequence which converges to a function of bounded variation.
There are other results in the paper which should have given Helly a much higher profile in the world of mathematics than he has achieved. One is the fact that he gives the Hahn-Banach theorem for the space C[a, b], while he is providing a simpler proof of a theorem which Riesz had published the previous year. He also gives the uniform boundedness principle for linear functionals, the Banach-Steinhaus theorem. As the authors of  write:-
Had Helly succeeded in staying in the mainstream of mathematics, as an academician who published and participated in seminars, he would have undoubtedly have capitalised on his earlier contributions. He not only might have seen to it that proper credit should be ascribed, but it is likely that he would have extended his results further. ... In most careers there are some disappointments and failures, but Helly's career derailed early, and life never gave him a chance to get back on the right track.
Article by: J J O'Connor and E F Robertson