Although he was born in the United States, Hille's parents were Swedish. When Einar Hille was two years old his mother returned to Sweden and lived in Stockholm. Hille would spend the next 24 years of his life in Sweden only returning to the United States when he was 26 years old. In 1900 Einar entered the Palmsgrenska Samskola, a private school in Stockholm. When he left in 1911 it was not mathematics in which he intended to make his career, rather it was chemistry.
Hille entered the University of Stockholm in 1911 with the intention of reading for a degree in chemistry. In fact he studied the subject for two years at university and he was taught by an exceptional chemist in Hans von Euler-Chelpin. He had been appointed professor of general and inorganic chemistry in 1906 and he worked on the biochemistry of sugar and phosphates and he was awarded the Nobel prize for chemistry in 1929 for his work on enzymes in the fermentation of sugar. He also helped determine the chemical structures of several vitamins. Hille made an impressive start to his career in chemistry with his first publication in 1913 being jointly with Euler-Chelpin. However, Hille decided that he did not have the necessary dexterity to make a career in a subject which involved delicate experiments. He therefore decided to give up his work with Euler-Chelpin and to study a topic which required no experimental expertise at all, namely mathematics.
Hille was fortunate in not only having a world class chemist to work under in Stockholm, but also world class mathematicians. He began to study under Ivar Bendixson and Helge von Koch. Hille was awarded his first degree in mathematics in 1913 and the equivalent of a Master's degree in the following year. Then Hille began working with Marcel Riesz on conformal mappings and submitted a thesis on that topic in 1916; for this he was awarded a Lic. Ph. During this time at Stockholm he was also strongly influenced by Mittag-Leffler.
By this time Europe was engulfed by World War I and Hille undertook war work, serving in the Swedish army for two years as a typist. While in the army he decided to undertake research for his Ph.D. and this he did without any further supervision. He received a Ph.D. from Stockholm in 1918 for a doctoral dissertation entitled Some Problems Concerning Spherical Harmonics. In 1919 Hille was awarded the Mittag-Leffler prize for his outstanding contributions, and was given the right to teach at the University of Stockholm. He was employed in the Swedish civil service for two years starting from the end of World War I.
Hille obtained a fellowship to work with Birkhoff in Harvard and he returned to the land of his birth in 1920, spending the academic year 1920-21 at Harvard where, as well as working with Birkhoff, he also studied with Kellogg. In academic year 1921-22 Hille was Benjamin Peirce Instructor at Harvard then, in 1922, he went to Princeton as an instructor.
In 1923 Hille was promoted to assistant professor at Princeton, and in 1927 to associate professor. In 1933, he was appointed full professor at Yale University and four years later, in 1937, he married Kirsti Ore, who was the sister of his Yale colleague, the mathematician Øystein Ore. Kirsti Hille wrote the article  after the death of her husband. Kirsti and Einar had two children, both sons. Hille was appointed director of graduate studies in 1938 and he held this position, and his chair, at Yale until he retired in 1962.
Hille's main work was on integral equations, differential equations, special functions, Dirichlet series and Fourier series. Later in his career his interests turned more towards functional analysis. Zund writes in :-
Hille was one of the few mathematicians who brought to his study of functional analysis - operator theory some twenty years experience in classical analysis. Moreover, he was almost unique among mathematicians in applying functional analysis to investigate classical problems, rather than simply considering abstract situations for their own sake.He is well known as an author of a number of texts. The first of these was Functional analysis and semigroups (1948). Stone, reviewing the book, writes:-
It is impossible in a review of this kind to give an adequate account of the wealth of material covered in the book, or to emphasize the interest of specific results. It is appropriate to note the systematic, incisive and polished character of the author's treatment. His scholarly handling of the subject will be appreciated by all users of the book, as will the clarity of his expository style.Among Hille's other texts were Analytic function theory Vol 1 (1959), Vol 2 (1964); Analysis Vol 1 (1964), Vol 2 (1966); Lectures on ordinary differential equations (1969); Methods in classical and functional analysis (1972); and Ordinary differential equations in the complex domain (1976). In the preface of Methods in classical and functional analysis Hille explains both about his aims in writing the text and his view of mathematics:-
Modes come and go in mathematics as in most fields. During the half-century and more that I have worked in the vineyard I have heard many dire predictions for the fate of my ideas and interests. Abstraction has been in the saddle during most of the time and has ridden us mercilessly. In a modest way I have taken part in this development. I did not believe in abstraction per se, one should know what one is trying to generalize and one should show that the generalization is significant. I have tried to keep at least one foot on the ground while craning my neck to look into Heaven. What is Heaven? There are some doubts, and the more extravagant claims of the abstract mathematicians to be the sole dispenser of the true faith and the arbiters of values are received with a healthy scepticism.... This book may be regarded as part of the backlash. If the book has a thesis, it is that a functional analyst is an analyst, first and foremost, and not a degenerate species of a topologist. His problems come from analysis and his results should throw light on analysis.... It seemed to me that I could do some useful work in giving the student a historical perspective and in showing how the multitude of abstract concepts have arisen and are present in Euclidean spaces.The article  list 175 papers and 12 books written by Hille. The book  reproduces 47 selected papers of Hille, published in the years 1922-1969. The book also contains a personal account of Hille's mathematical career which he gave at the Yale Colloquium in May 1962. The same autobiographical paper appears as .
Hille served as president of the American Mathematical Society (1937-38) and was the Society's Colloquium lecturer in 1944. As well as editorial work with the Annals of Mathematics (1929-33) and the Transactions of the American Mathematical Society (1937-43) he received many honours including election to the National Academy of Sciences (1953) and the Royal Academy of Sciences of Stockholm. He was honoured by Sweden with their award of the Order of the North Star.
Article by: J J O'Connor and E F Robertson