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A () is a real vector space together with a mapping : × → ℝ satisfying for all ∈ and λ ∈ ℝ. Concerning the notation : × → ℝ and others we shall use later on, see the section of this book. Instead of () we will write or, occasionally, . The laws above are then the following: for all ∈ , λ ∈ ℝ, and := > 0 for all ∈ {0}. Instead of () we mostly will speak of , hence tacitly assuming that is equipped with a fixed , i.e. with a fixed : → ℝ satisfying rules (i), (ii), (iii), (iv).

designates again an arbitrary real inner product space containing two linearly independent elements. As throughout the whole book, we do not exclude the case that there exists an infinite and linearly independent subset of .

Also in this chapter denotes a real inner product space of arbitrary (finite or infinite) dimension ≥ 2.

As in the chapters before, denotes a real inner product space of arbitrary (finite or infinite) dimension ≥ 2.

Let () and () be arbitrary real inner product spaces each containing at least two linearly independent elements. However, as in the earlier chapters we do not exclude the case that there exist infinite linearly independent subsets of or . One of the important results of this chapter is that the hyperbolic geometries (()), (()) over = (), = (), respectively, the group of hyperbolic motions, are isomorphic (see p. 16f) if, and only if, () and () are isomorphic (see p. 1f).