M(?)bius Transformations In Hilbert Spaces

The first chapter of this paper is some basic facts relate to the subjects we study, including the definition of M(o|¨)bius transformations and its basic propertics;In the second chapter we extend the usual non-Euclidean metric to the non-Euclidean metric in unit ball of Hilbert space,and prove that M(B),the group of M(o|¨)bius transformations keeping the unit ball,is exactly the isometric group with respect to this metric;In the third chapter we give a complete classification of M(o|¨)bius transformations in Hilbert space and discuss related properties.