In this paper, we mainly study the isometry groups in infinite dimensions innerproduct spaces and the discrete Mo|¨bius groups in higher dimensions. this thesis isarranged as follows:In chapter 1, we provide some background of our problems which will be intro-duced and the statement of our main results.In chapter 2, we discuss the isometry groups consists of Mo|¨bius transformationswhich mapping the unit ball B onto B in inner product spaces: First, we investigatethe relationship between the re?ections and the Mo|¨bius transformations, we also gainone necessary and su?cient condition for a map to be a Mo|¨bius transformations ininner product spaces, thus we have some conclusions like in Euclidean spaces; Next, wegive the precise description about equicontinuity on Mo|¨bius transformations in innerproduct spaces. Last, we establish the Jφgensen inequality in special case by usingpure algebraic method.In chapter 3, we discuss the discrete Mo|¨bius groups in higher dimensions, by us-ing the chordal norm and Cli?ord matrices we obtain three necessary condition of thediscrete groups which are generated by Mo|¨bius transformations in higher dimensionsEuclidean spaces .
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