A Representation of Hyperbolic Motions Including the Infinite-Dimensional Case

Let X be a real inner product space of (finite or infinite) dimension ???2, O(X) be its group of all surjective (hence bijective) orthogonal transformations of X, T(X) be the set of all hyperbolic translations of X and M(X, hyp) be the group of all hyperbolic motions of X. The following theorem will be proved in this note. Every ${\mu\in M(X,{\mbox hyp})}$ has a representation ?? = T · ?? with uniquely determined ${T\in T(X)}$ and uniquely determined ${\omega\in O(X)}$ .