On M(?)bius Transformations In Infinite Dimension

In this paper, we mainly study Mobius transformations in infinite dimension. A series of new results are obtained.In chapter 1, the background of our interested problems is introduced and our main results are stated.In chapter 2, we give the related preliminaries of Mobius transformations in infinite dimension, and by using Clifford matrices we obtain some elementary properties of Mobius transformations in infinite dimension. Futhermore, we give the classification of Mobius transformations in infinite dimension and define two special kinds of elements, namely, hyperbolic and strictly parabolic elements, and establish their corresponding type criteria.In chapter 3, we focuse on the ball-preserving Mobius transformations in infinite dimension. After introducing the hyperbolic and chordal metrics, we get two necessary and sufficient conditions, one for the ball-preserving property of Mobius transformations in infinite dimension and the other for being isometries in infinitely dimensional unit ball with regard to the hyperbolic metric. We also gain a distortion theorem which shows the relationship between the hyperbolic metric and the chordal metric.In chapter 4, we discuss the condition for a two-generator subgroup to be discrete and nonelementary, and establish special forms of JΦrgensen's inequality in infinite dimension. At last we generalise some of the inequalities about the chordal metric to the case of infinite dimension.