Topological Vector Spaces Chapters 15

The task of model theory is to investigate mathematical structures with the aid of formal languages. Classical model theory deals with algebraic struc- tures. Topological model theory investigates topological structures. A to- pological structure is a pair (=,a) consisting of an algebraic structure ~ and a topology ~ on A. Topological groups and topological vector spaces are examples. The formal language in the study of topological structures is L t- This is the fragment of the (monadic) second-order language (the set variab- les ranging over the topology ~) obtained by allowing quantification over set variables in the form 3X(t e X ^ ~), wheret is a term and the second- order variable X occurs only negatively in ~ (and dually for the universal quantifier). Intuitively, L t allows only quantifications over sufficiently small neighborhoods of a point.

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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 (" ~O- forms") . Certain among the resul ts 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 ~ -dimensional 0 spaces ideally serves the purpose. First, these spaces show a large nurober of phenomena typical of infinite dimensional spaces.