He chose to attend the École Polytechnique and he while still an undergraduate there published his first paper on semiconvergent series in 1905. After graduating in first place, Lévy took a year doing military service before entering the École des Mines in 1907. While he studied at the École des Mines he also attended courses at the Sorbonne given by Darboux and Émile Picard. In addition he attended lectures at the Collège de France by Georges Humbert and Hadamard.
It was Hadamard who was the major influence in determining the topics on which Lévy would undertake research. Finishing his studies at the École des Mines in 1910 he began research in functional analysis. His thesis on this topic was examined by Émile Picard, Poincaré and Hadamard in 1911 and he received his Docteur ès Sciences in 1912.
Lévy became professor École des Mines in Paris in 1913, then professor of analysis at the École Polytechnique in Paris in 1920 where he remained until he retired in 1959. During World War I Lévy served in the artillery and was involved in using his mathematical skills in solving problems concerning defence against attacks from the air. A young mathematician R Gateaux was killed near the beginning of the war and Hadamard asked Lévy to prepare Gateaux's work for publication. He did this but he did not stop at writing up Gateaux's results, rather he took Gateaux's ideas and developed them further publishing the material after the war had ended in 1919.
As we indicated above Lévy first worked on functional analysis :-
... done in the spirit of Volterra. This involved extending the calculus of functions of a real variable to spaces where the points are curves, surfaces, sequences or functions.In 1919 Lévy was asked to give three lectures at the École Polytechnique on (see ):-
... notions of calculus of probabilities and the role of Gaussian law in the theory of errors.Taylor writes in :-
At that time there was no mathematical theory of probability - only a collection of small computational problems. Now it is a fully-fledged branch of mathematics using techniques from all branches of modern analysis and making its own contribution of ideas, problems, results and useful machinery to be applied elsewhere. If there is one person who has influenced the establishment and growth of probability theory more than any other, that person must be Paul Lévy.Loève, in , gives a very colourful description of Lévy's contributions:-
Paul Lévy was a painter in the probabilistic world. Like the very great painting geniuses, his palette was his own and his paintings transmuted forever our vision of reality. ... His three main, somewhat overlapping, periods were: the limit laws period, the great period of additive processes and of martingales painted in pathtime colours, and the Brownian pathfinder period.Not only did Lévy contribute to probability and functional analysis but he also worked on partial differential equations and series. In 1926 he extended Laplace transforms to broader function classes. He undertook a large-scale work on generalised differential equations in functional derivatives. He also studied geometry.
His main books are Leçons d'analyse fonctionnelle Ⓣ (1922), Calcul des probabilités Ⓣ (1925), Théorie de l'addition des variables aléatoires Ⓣ (1937-54), and Processus stochastiques et mouvement brownien Ⓣ (1948).
In 1963 Lévy was elected to honorary membership of the London Mathematical Society. In the following year he was elected to the Académie des Sciences.
Loève sums up his article  in these words:-
He was a very modest man while believing fully in the power of rational thought. ... whenever I pass by the Luxembourg gardens, I still see us there strolling, sitting in the sun on a bench; I still hear him speaking carefully his thoughts. I have known a great man.
Article by: J J O'Connor and E F Robertson
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