Nina Bari's parents were Olga Eduardovna Seligson and Karl Adolfovich Bari. Karl Adolfovich was a medical doctor. Nina Karlovna was educated at a private school, L O Vyazemska's High School for Girls in Moscow, where she showed great potential in mathematics. At this time the education available for girls in Russia was of lower quality than that of boys and, as a consequence, the final examinations for girls were of a lower standard than those for boys. However, Bari wanted to show her proficiency in mathematics so she took the highly unusual step of taking the boy's leaving examinations. After the October 1917 Revolution in Russia the Bolshevik Party introduced major educational reforms, the most relevant to Bari being that for the first time universities were open to women as well as men. Before the Revolution, women who wanted further education attended colleges such as the Moscow Women's College. The year 1918 was the first in which women were allowed to enter Moscow State University and this coincided nicely with Bari achieving the entrance qualifications. In fact the University had been closed during the disruption caused by the Revolution and 1918 was the first year in which it reopened. She entered the Faculty of Mathematics and Physics at Moscow State University at a time when it was becoming an extremely vigorous research environment.
In the Moscow School of Mathematics she came under the influence of Nikolai Nikolaevich Luzin but he was only one of a number of world-class mathematicians teaching at the university at this time, including Sergei Alekseevich Chaplygin, Dimitri Fedorovich Egorov, Vyacheslaw Vassilievich Stepanov and Nikolai Egorovich Zhukovsky. Among the undergraduate students Pavel Samuilovich Urysohn was in his final year of study and remained part of the School of Mathematics undertaking research. This strong mathematical group, which the students called 'Luzitania', was joined by Pavel Sergeevich Aleksandrov in 1920 :-
Nina Bari's student years coincided with the rapid growth of the Moscow Mathematical school, which was to be a very great influence in the subsequent blossoming of the whole of mathematics in the Soviet Union. This was the period of intensive development of the Moscow real variable school, headed by N N Luzin, under whose guidance Nina Bari started her own mathematical work while still an undergraduate. In fact, it was then that she first began to study the uniqueness problem for trigonometric series.
To see why 'Luzitania' was so successful, we need look no further than the comments that Aleksandrov made about this time:-
To see Luzin in those years was to see a display of what is called an inspired relationship to science. I learnt not only mathematics from him, I received also a lesson in what makes a true scholar and what a university professor can and should be. Then, too, I saw that the pursuit of science and the training of young people in it are two facets of one and the same activity - that of a scholar.
Bari graduated in 1921, and having completed the four-year course in only three years, she became the first woman to graduate from Moscow State University. She began teaching mathematics at a number of institutions, the Moscow Forestry Institute, the Moscow Polytechnic Institute, and the Sverdlov Communist Institute. However, her aim was to become a university teacher and having sampled the excitement of 'Luzitania' while an undergraduate she applied for a research fellowship. She won one of a small number of available scholarships and joined the Research Institute of Mechanics and Mathematics at Moscow State University. Bari began research at the Institute, with Luzin as her thesis advisor, but continued to hold her teaching posts. She undertook research on the theory of trigonometrical series and her major results were announced in her first paper Sur l'unicite du developpement trigonometrique Ⓣ published by the Académie des Sciences in Paris in 1923. In the same year she presented the results in a lecture to the Moscow Mathematical Society (becoming the first woman to address the Society). In addition to support from Luzin, she was strongly influenced by Dmitrii Evgenevich Menshov who had undertaken research with Luzin but returned to Moscow State University as a lecturer in 1922. Bari's thesis On the uniqueness of trigonometric expansions was submitted in 1925 and the degree awarded after she defended her thesis in January 1926. The exceptional quality of this thesis led to her being awarded the Glavnauk Prize. After this Bari became a research assistant at the Institute of Mathematics and Mechanics in Moscow. She published further papers in 1926, namely Sur la representation analytique d'une classe de fonctions continues Ⓣ and (with D E Menshov) Sur l'integrale de Lebesgue-Stieltjes et les functions absolument continues de fonctions absolument continues Ⓣ both of which had the distinction of also appearing in the Académie des Sciences publications. In 1927 she published detailed proofs of the results of her thesis which she had announced in her first paper :-
Already this first piece of work by Nina Bari testified to her great mathematical talent, since it included the solution of several very difficult problems in the theory of trigonometric series that had lately been engaging the attention of many outstanding mathematicians.
During 1927-29 she spent time at the Sorbonne and the Collège de France in Paris, attending lectures by Jacques Hadamard. After attending the Polish Mathematical Congress in Lvov, she also attended the International Congress of Mathematicians in Bologna in 1928 at which she gave the invited lecture Sur la structure analytique d'une fonction continue arbitraire Ⓣ. Following this, she was awarded a Rockefeller fellowship which funded a second year-long visit to Paris. In 1929 she returned to Moscow State University where she became a full professor in 1932. Three years later she was awarded the equivalent of a D.Sc. in Physical-Mathematical Sciences. This degree, normally conferred after submitting a thesis, was awarded to Bari without the thesis requirement :-
... since she was already recognised as one of the leading mathematicians specializing in the theory of functions of a real variable.
The year Bari graduated from Moscow State University, Viktor Vladmirovich Nemytski entered the university to read mathematics. They became close friends sharing not only mathematical interests but also a love of hiking in the mountains. They were eventually married. The vigorous school at Moscow State University headed by Luzin began to run out of steam in the last few years of the 1920s as he concentrated on writing monographs. He left the University in the early 1930s to work at the USSR Academy of Sciences and from 1935 on he led the Department of the Theory of Functions of Real Variables at the Steklov Institute. This left Bari and Menshov to take over his role of leading the School of Function Theory which they did in the 1940s.
Bari was an outstanding research mathematician who wrote over fifty research articles. The authors of  write:-
Nina Bari was one of the acknowledged leaders of the Soviet real variable school, and a worthy representative of this discipline at the Moscow University with which it is so closely associated. Her numerous papers exerted a great influence on the development of such fundamental branches of the theory of functions as the theory of trigonometric series, orthogonal series and bases, etc. Several of her investigations are rightly regarded as classics, for example her papers on the theory of uniqueness for trigonometric series and on the superpositions of functions. ... a number of her papers, for example those on the theory of uniqueness and on bases, have served as starting-points for new lines of research in the theory of functions, afterwards intensively developed ...
Let us mention in particular her paper The uniqueness problem of the representation of functions by trigonometric series (Russian) which she published in 1949. Antoni Zygmund describes it as:-
... an exhaustive review, in many cases accompanied by complete proofs, of the existing results in the theory of uniqueness of representation of functions by trigonometric series.
This paper was translated into English and was published as a 90-page book by the American Mathematical Society in 1951. In  her final publication, a 936-page research monograph on trigonometric series, is described as follows:-
The range and depth of topics covered is quite extensive, and most of her work in the field is included. But even within so long a monograph, the subject could not be completely exhausted. ... It has become a standard reference for mathematicians specializing in the theory of functions and the theory of trigonometric series.
The fifteen chapters of the book are: Basic concepts and theorems; Fourier coefficients; Convergence of a Fourier series at a point; Fourier series of continuous functions; Convergence and divergence of a Fourier series on a set; "Correcting" a function on a set of small measure; Summability of Fourier series; Conjugate trigonometric series; Absolute convergence of Fourier series; Sine and cosine series with decreasing coefficients; Lacunary series; Convergence and divergence of general trigonometric series; Absolute convergence of general trigonometric series; The problem of uniqueness of the expansion of a function in a trigonometric series; and Representation of functions by trigonometric series.
Bari also wrote textbooks for use in teaching training institutes such as Higher Algebra (1932) and The Theory of Series (1936). She edited the complete works of Luzin and was the editor of two important mathematics journals. She also translated Lebesgue's famous book on integration into Russian.
She died by falling in front of a train on the Moscow Metro. It has been claimed that this was suicide due to depression caused by Luzin's death eleven years earlier. One of her students, P L Ul'yanov, wrote after her death :-
The untimely death of N K Bari is a great loss for soviet mathematics and a great misfortune for all who knew her. The image of Bari as a lively, straightforward person with an inexhaustible reserve of cheerfulness will remain forever in the hearths of all who knew her.
Finally let us give some indication of her personality and interests outside mathematics. We have already mentioned her love of hiking which took her into seriously difficult terrains in the Caucausus, Pamir and Tian Shan mountains. Her love of music and the arts included ballet, literature and poetry. Mikhail Lavrent'ev and L A Lyusternik write of her :-
... lively temperament, directness, with an inexhaustible supply of young vigour ...
Article by: J J O'Connor and E F Robertson