He studied under the greatest rabbis of his generation, including such legendary figures as the Sfas Emes, Rabbi Meir Simcha of Dvinsk and the Ridvaz, as well as learning under and with the Avnei Nezer; Rabbi Chaim Soloveitchik; the Rogatchover Gaon; the Chofetz Chaim; and Rabbi Aharon Kotler, all world-renowned rabbinic luminaries. Rabbi Greenspan obtained semicha at the young age of eighteen, indicative of his acceptance into the highest circles of rabbinic scholarship due to his tremendous knowledge and ability.
Rabbi Greenspan died at the age of eighty-three, and with his death Europe lost one of its primary and most exceptional Torah scholars. A large number of people attended his funeral, where he was eulogised by Chief Rabbi Israel Brodie, Rabbi Leib Gurwicz, Dayan Grosnass of the London Beth Din and Rabbi Eliezer Lopian, rosh yeshiva of Toras Emes in London.
Writing of Rabbi Greenspan in the Jewish Chronicle, Rabbi Nosson Ordman described his deceased colleague as the "humblest and most well-mannered of men" despite his "scholarship and intellectual" abilities. He noted that Rabbi Greenspan's death was "a great and irreplaceable loss both to the yeshiva and Anglo-Jewry. He was... one of the outstanding... Talmudists of this generation". Rabbi Jakobovits, in deep reveration of Rabbi Greenspan, commented, "he may have been the greatest Torah scholar in England" and also noted his teacher's great modesty and knowledge of secular disciplines .