In becoming a student of Schwartz, Lions was being supervised by an outstanding mathematician who had the previous year won a Fields medal. Schwartz had received the medal at the International Congress of Mathematicians in Harvard on 30 August 1950 for his work on the theory of distributions.
Sa pagiging isang mag-aaral ng Schwartz, Lions ay ini-supervised sa pamamagitan ng isang natitirang dalubbilang na ang nakaraang taon napanalunan ng isang patlang na medalya.