A Representation of Hyperbolic Motions Including the Infinite-Dimensional Case

Abstract

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 μ = T · ω with uniquely determined \({T\in T(X)}\) and uniquely determined \({\omega\in O(X)}\) .