This is an easy consequence of the Schwarz lemma. As a corollary of the theorem, any two simply connected open subsets of the Riemann sphere which both lack at least two points of the sphere can be conformally mapped into each other. ==History== The theorem was stated (under the assumption that the boundary of U is piecewise smooth) by Bernhard Riemann in 1851 in his PhD thesis.
However, in order to be valid, the Dirichlet principle needs certain hypotheses concerning the boundary of U which are not valid for simply connected domains in general. The first rigorous proof of the theorem was given by William Fogg Osgood in 1900.
He proved the existence of Green's function on arbitrary simply connected domains other than C itself; this established the Riemann mapping theorem. Constantin Carathéodory gave another proof of the theorem in 1912, which was the first to rely purely on the methods of function theory rather than potential theory.
Carathéodory continued in 1913 by resolving the additional question of whether the Riemann mapping between the domains can be extended to a homeomorphism of the boundaries (see Carathéodory's theorem). Carathéodory's proof used Riemann surfaces and it was simplified by Paul Koebe two years later in a way that did not require them.
Another proof, due to Lipót Fejér and to Frigyes Riesz, was published in 1922 and it was rather shorter than the previous ones.
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