For about a decade I have made an effort to study quadratic forms in
infinite dimensional vector spaces over arbitrary division rings. Here
we present in a systematic fashion half of the results found du- ring
this period, to wit, the results on denumerably infinite spaces ("
NO-forms'''). Certain among the results included here had of course been
published at the time when they were found, others appear for the first
time (the case, for example, in Chapters IX, X, XII where I in- clude
results contained in the Ph.D.theses by my students W. Allenspach, L.
Brand, U. Schneider, M. Studer). If one wants to give an introduction to
the geometric algebra of infinite dimensional quadratic spaces, a
discussion of N-dimensional O spaces ideally serves the purpose. First,
these spaces show a large number of phenomena typical of infinite
dimensional spaces. Second, most proofs can be done by recursion which
resembles the familiar pro- cedure by induction in the finite
dimensional situation. Third, the student acquires a good feeling for
the linear algebra in infinite di- mensions because it is impossible to
camouflage problems by topological expedients (in dimension NO it is
easy to see, in a given case, wheth- er topological language is
appropriate or not).