A fundamental theorem for dimension-free Möbius sphere geometries

Summary.

Let (X, δ) and (V, ε) be real inner product spaces of (finite or infinite) dimensions dim X, dim V greater than 1 (see our book [1] for special notions, results and the notation applied in the present paper). Especially the following Theorem 2 will be proved. The Möbius sphere geometries $(X \bigcup \{\infty \},{\mathbb{M}}(X, \delta)), (V \bigcup \{\infty \},{\mathbb{M}}(V, \epsilon))$ over (X, δ), (V, ε), respectively, where ${\mathbb{M}}$ is the Möbius group, are isomorphic (see [1], p. 16 f) if, and only if, $(X, \delta) \cong (V, \epsilon)$ (see [1], p. 1 f).