Extensions of distance preserving mappings in euclidean and hyperbolic geometry

Suppose that X is a real inner product space of (finite or infinite) dimension at least 2. A distance preserving mapping, where is a (finite or infinite) subset of afinite-dimensional subspace of X, can be extendedto an isometry of X. This holds true foreuclidean as well as for hyperbolic geometry. To both geometries there exist examplesof non-extentable distance preserving , where Sis not contained in a finite-dimensional subspace ofX.