Editing Uniformization theorem (section)

===Hilbert space methods===
{{See also|Planar Riemann surface#Uniformization theorem}}
In 1913 Hermann Weyl published his classic textbook "Die Idee der Riemannschen Fläche" based on his Göttingen lectures from 1911 to 1912. It was the first book to present the theory of Riemann surfaces in a modern setting and through its three editions has remained influential. Dedicated to [[Felix Klein]], the first edition incorporated [[David Hilbert|Hilbert's]] treatment of the [[Dirichlet problem]] using [[Hilbert space]] techniques; [[L. E. J. Brouwer|Brouwer's]] contributions to topology; and [[Paul Koebe|Koebe's]] proof of the uniformization theorem and its subsequent improvements. Much later {{harvtxt|Weyl|1940}} developed his method of orthogonal projection which gave a streamlined approach to the Dirichlet problem, also based on Hilbert space; that theory, which included [[Weyl's lemma (Laplace equation)|Weyl's lemma]] on [[elliptic regularity]], was related to [[W. V. D. Hodge|Hodge's]] [[Hodge theory|theory of harmonic integrals]]; and both theories were subsumed into the modern theory of [[elliptic operator]]s and {{math|''L''<sup>2</sup>}} [[Sobolev space]]s. In the third edition of his book from 1955, translated into English in {{harvtxt|Weyl|1964}}, Weyl adopted the modern definition of differential manifold, in preference to [[triangulation (topology)|triangulations]], but decided not to make use of his method of orthogonal projection. {{harvtxt|Springer|1957}} followed Weyl's account of the uniformisation theorem, but used the method of orthogonal projection to treat the Dirichlet problem. {{harvtxt|Kodaira|2007}} describes the approach in Weyl's book and also how to shorten it using the method of orthogonal projection. A related account can be found in {{harvtxt|Donaldson|2011}}.