Editing Maximal compact subgroup (section)

===Proofs===
For a real semisimple Lie group, Cartan's proof of the existence and uniqueness of a maximal compact subgroup can be found in {{harvtxt|Borel|1950}} and {{harvtxt|Helgason|1978}}. {{harvtxt|Cartier|1955}} and {{harvtxt|Hochschild|1965}} discuss the extension to connected Lie groups and connected locally compact groups.

For semisimple groups, existence is a consequence of the existence of a compact [[Complexification (Lie group)|real form]] of the noncompact semisimple Lie group and the corresponding [[Cartan decomposition]]. The proof of uniqueness relies on the fact that the corresponding [[Riemannian symmetric space]] ''G''/''K''  has [[negative curvature]] and 
Cartan's fixed point theorem. {{harvtxt|Mostow|1955}} showed that the derivative of the exponential map at any point of ''G''/''K'' satisfies |d exp ''X''| ≥ |X|. This implies that ''G''/''K'' is a [[Hadamard space]], i.e. a [[complete metric space]] satisfying a weakened form of the parallelogram rule in a Euclidean space. Uniqueness can then be deduced from the [[Bruhat-Tits fixed point theorem]]. Indeed, any bounded closed set in a Hadamard space is contained in a unique smallest closed ball, the center of which is called its [[circumcenter]]. In particular a compact group acting by isometries must fix the circumcenter of each of its orbits.