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topology as the norm.
==Other holomorphic fixed point theorems==
In finite dimensions the existence of a fixed point can often be deduced from the [[Brouwer fixed point theorem]] without any appeal to holomorphicity of the mapping. In the case of [[bounded symmetric ___domain]]s with the [[Bergman metric]], {{harvtxt|Neretin|1996}} and {{harvtxt|Clerc|1999}} showed that the same scheme of proof as that used in the Earle-Hamilton theorem applies. The bounded symmetric ___domain ''D'' = ''G'' / ''K'' is a complete metric space for the Bergman metric. The open semigroup of the complexification ''G''<sub>''c''</sub> taking the closure of ''D'' into ''D'' acts by [[contraction mapping]]s, so again the Banach fixed-point theorem can be applied. Neretin extended this argument by continuity to some infinite-dimensional bounded symmetric domains, in particular the Siegel generalized disk of symmetric Hilbert-Schmidt operators with operator norm less than 1. The
==References==
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