Having completed his studies in law, science and engineering, Salem then went into banking and started working for Banque de Paris et des Pays-Bas in 1921.

In the spring of 1939, Salem collaborated with the young Polish mathematician Józef Marcinkiewicz, and he continued to write mathematics papers while working for the bank.

With a deteriorating political situation in France and the beginning of World War II in September 1939, Salem was called up for military duty.

In 1967, Éditions Hermann published Salem's Oeuvres mathématiques, edited by his collaborators Antoni Zygmund and Jean-Pierre Kahane.

Salem's older son survived the war and he enlisted in the free French Forces and took part in Allied landings in the South of France in 1944.