No one would assert that finite dimensionality is an intrinsic feature of the concept of quadratic form. Yet, apart from a very small number of results (see References to Chapter XI) there has been, as far as we know, only Kaplansky’s 1950 paper on infinite dimensional spaces pointing our way, namely in the direction of a purely algebraic theory of quadratic forms on infinite dimensional vector spaces over “arbitrary” division rings. Such a theory would, naturally, leave aside the highly developed theory of Hilbert spaces and its relatives, Krein spaces, Pontrjagin spaces (see  for an orientation on these topics). Furthermore, when we speak of infinite dimensional geometric algebra we do not, in this book, mean discussion of the ramifications into geometry of hypotheses belonging to set theory nor, reversely, the study of axioms forced upon set theory by geometry. We simply mean that the (algebraic) dimension of the quadratic spaces is allowed to be infinite. Many problems of the finite dimensional setting remain perfectly meaningful and invite an investigation when the finiteness condition is removed. Our results show that it is possible to generalize, without rarefying, classical results from finite dimensional orthogonal geometry.
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