Following work of Hardy, Beurling found an attractive formulation of the uncertainty principle which he never published. It appeared among his unpublished notes, and the statement was supplied in the Collected Works of Beurling. Then around 1993, Hörmander published a proof which he had heard directly from Beurling in Princeton. We now supply a different and rather elementary proof, which allows substantial reduction in the assumptions. The seminar is based on an article the text of which is in http://www.math.kth.se/~haakanh/bibliography.html